you guys we are part six is this part six part six y'all gray skies and it is getting juicy yeah excuse the back here my baby has been sleeping in girl been sleeping in my room at night cuz he's so big you guys we got to get him either a full bed or a queen size bed my baby is big JB sits at like four six four seven and baby boy weighs like 85 86 pounds and he's a little chubby but he's a big kid he's tall but he's still my baby he just turned eight he just turned 80 almost 90 pounds but he's tall y'all I'll have to stand up in one of my blocks again you'll see how tall my child is to me so anyway girl let's get into this story time let me kind of turn this light down something but I just wanted to explain if you see things a little bit of a disarray in the back okay you guys so we ended off the last story time with Kendra and Lamar arguing Lamont could be Lamar too Kendra and Lamont arguing heading into the surprise baby shower that she has no idea about she you know she's arguing like I need to know what's going on I feel like you with holding something from me you know what's going on Lamont you know turns the corner and she sees there literally her best friend Tiffany her parents Aunt Nora Aunt Ella petty ass even barman is there a couple of their close basically their close friends and families have gathered together flew out from Chicago for this surprise party you know the surprise party Kendra starts bawling and that's when like I said Tiffany runs to her and she escorts her to one of the bathrooms and you know can't Tiffany's trying to calm her down she's like Kendra are you okay and she's like yeah I just I just made a fool of myself I can't believe that just happened and so Tiffany's like what's going on why were you guys arguing like that and so Kendra's like Tiff I don't know just lately I just felt like we have him in communicating and you know when he came back from you know his trip he was secretive and he basically liked about being on the phone and Tiffany was like he was on the phone with me we were planning this you know we were busy planning this baby shower and she looks at her she's like are you serious like yes Kendra and so that's when Tiffany hugs her she's like it's okay it's okay and so Kendra kind of calms down when she goes back into the restaurant she can hear people whispering right and she passed by one of her friends and child one of her friends and Aunt Ella whispered she's like yeah that's the Byron right there and Byron is sitting at a table with his Aunt Nora that's N-O-R-A y'all my accent Byron is sitting at a table with Aunt Nora and he just smiles at her you know I'm pretty sure he hears some of that too so anyway Kendra says hi to everyone and she embraces her parents and her mom was like Kendra are you okay she's like yeah mama I'm fine she's like you know I didn't know things were bad with you guys she said it's it's not bad it was just a big misunderstanding and I'm okay I'm fine parents end up deciding to stay with them for the weekend and so the party goes well you guys Kendra gets a lot of presents and she's so happy to see some of her family there he played a couple of games they ate and then they ended up leaving again her parents do decide to stay with them for the weekend okay so this is on a Friday night and so that night they go upstairs and Kendra again she's tired she sits down and she looks at Lamont and she could tell Lamont is upset he's embarrassed and he's quiet again she's like Lamont I am so sorry I just thought that you know he said Kendra look if I'm trying to explain to you what I just put me for you to hear me out you know I've been telling you all this time that nothing was going on and I feel like you don't believe you weren't believing me so if anything I should be questioning you now that I know that you had Byron to go with you on the doctor's appointment to look I shouldn't have said that I didn't mean anything by he said so basically you said that to hurt me or why why would you was it not true they know it was true he did come to the doctor's appointment but it's not what you think and so he's like look I'm tired let's just go to bed your parents are downstairs I'm just gonna go downstairs to get some work done and I'll just come back up and she looks at him feeling guilty and she's like okay man so she gets into bed Lamont go downstairs he goes he gets to his office y'all he pulls up his desktop and he had the hours before he had went ahead online and submitted a thousand dollars for a bed online and he saw he had doubled his money he had won $2000 and he's feeling excited and I look I don't know anything about gambling so we're just gonna say it's fake slots or something so he figured $2000 let me try it again so he goes ahead and bids he's like should I do all of it or have he's like screw it all of it he goes ahead and bid all his money again closes his laptop and goes upstairs so y'all he's back to gambling you know so the weekend goes off without a hit you know Kendra's mom and dad are happy to be there and Kendra is you know she's reassures her parents again that it was a big mother big misunderstanding and that you know things are well with her and Lamont now we're gonna kick over to Aunt Ella child do I have a phone actually hold on y'all let me go get Jamie's Christmas present here's Aunt Ella child here's Aunt Ella yeah excuse me I got this old-school phone my baby's phone don't ask her out here's not Ella on the phone gossiping mind you why she's on the phone this is the next day okay she's on the phone and mind you what's next gonna come in on her Lamont's gonna walk in on her mm-hmm yeah yes child well apparently she had that young boy to go to that appointment with her knowing that my baby was on on a business trip can you believe that yeah mm-hmm hold on oh honey my Lamont hold on Lamont Lamont is out here at the front door so child Lamont I walked in he's like Aunt Ella who are you talking to she said nobody baby he's like on Ella who she said yeah just one of the church members that's all so are you in Kendra okay he's like on Ella yeah we okay she's a little bit y'all arguing about he's an Ella was a huge misunderstanding between Kendra and I so Lamont head came over because I'm Ella was having problems with her car Lamont was gonna look at her car okay look at the transmission or something wrong he goes out to look at her at her car this is an Ella hello child he on went outside to work on my car yeah girl I think it's my carburetor or something yeah like I said I don't know if that young boy is gonna cause some problems but I know Lamont better get his wife together mm-hmm child old people just as petty old people just as petty as hell so Aunt Ella's gossiping with other church members a lot about what happened at the party yeah this is a small town so it gets around so it's a couple of days go by y'all and this is Lamont talking to Kendra he's like you know what um I don't think we need Byron to be around as much anymore and Kendra's like what do you mean you know I'm not even I'm just halfway through my pregnancy and we have so much more to do I gotta get the nursery ready together and so I can help you with that it's just like Lamont you but you're always in your office busy and so you're gonna help me to put up the drapes so you're gonna help me to unpack we got to you know get the basement together he's like well how about this how about you know I go ahead and help you out a little bit more and that way he only needs to come by one to two days out of the week instead of three to four and she's like okay was but since this is your idea you need to let him know and so Kendra continues to eat and she's like hmm he's a y'all can do that he's a yeah no problem I can do that so the next day Lamont goes up to Byron he's like hey Byron man you got a second I want to talk to you about something and Byron turns around looking all fine he's like yeah man what's up what's up he's like well look you know I scored this business deal a couple of months ago that will allow me to be home a little bit more with Kendra so we really only need you to come by now only you know one or two days out of the week instead of three or four days and Byron kind of looks at him he's like are you sure he's like yeah I'm sure and so by honest like okay then so Lamont turns to leave but then Byron was like hey hold up he's in Lamont turns around like shit and so he's a look man to man I just want to ask you something you know you cutting back my hours that doesn't have anything to do with that fight that I heard you and Kendra arguing about right Lamont looks at him like this because that's what Lamont does he's like no man it has nothing to do about that you know we have both agreed once she was halfway through the pregnancy is it and again if I got this you know this is still that I would be able to help her out a lot more and so Byron looks at him and Byron knows he's lying he knows that that's only half truth and so he's like alright man no problem so he continues to work Lamont goes in feeling feeling like Lamont like he's achieved some stuff he goes into his office he sees that he had now won the next gambling thing and y'all this is how it works you figure that you can win everything everything and you gamble more and more and more and so he's back until he's been sucked into that habit again all right you guys so Kendra comes in to the office knocks on the door goes in she's like so did you speak to him he's like yeah he was fine with it and Kendra kind of looks at him she's like okay then fine so Kendra walks out and she goes downstairs she goes out into the back to see about it she's okay fine and so Kendra was like it seems like your aunt is doing well you know is everything okay with her he's like yeah I mean sometimes I think she's okay but I know that you know I know that deep down she's not and so I still am going to have to look into finding somewhere for her to be at he's actually I was wondering if I could ask you for a favor and it actually has to do with my aunt Nora and she's like yeah sure what is it and he's like yeah my aunt kind of took a liking to you so I was wondering if you would mind coming with me to look at some of these nursing homes that I'm thinking about putting her in and Kendra kind of smells she's like absolutely whatever we can do to help you just let me know when you want to go you know and I'll be more than happy to you know visit these places man he's like thank you I really appreciate that so alright you guys so Kendra agrees so now let's cut over to aunt Nora and she's climbing up to bed next to her husband look at his husband her husband me what's a good old old southern black man name Glenn uncle Glenn yeah yeah that's a good man so this is Aunt Ella Aunt Ella would not let go of this argument that they heard between Kendra and Lamont and so here he go here she go you know I'm kind of what weren't Lamont about these city girls you know they fast and I understand Kendra you know as a godfair woman but you gotta have open communication and that's when Uncle Glenn put down his newspaper whatever the hell he was reading it and bitch he's a yeah communication like us so she looked in him she's like now look here Glenn now I'm not in the mood to be going back and forth with you but you know what I mean these young couples they get in marriage you know man just hard work you know you gotta work at it she slips into bed and so that's when Uncle Glenn said are you ever gonna tell Lamont the truth then since we talked about communication and she looked over at Glenn Uncle Glenn and she said I'll tell him and do time honey I'll tell him all right you guys got to figure out what secret is on


Hello all, I'm Shreya Rangini, working at Samsung Semiconductor India R&D as a embedded system engineer and primarily responsible for the Ethernet IP driver development. Before we start our presentation, let me tell you a short story. Initially when I started working on Ethernet, there was a situation where I need to know the basic informations of the network interface. I started using if config. I was aware that if config gives some basic information, while if config is good for some basic information about network interface, but for some advanced operations and information, we need to have something better than this. It was then while exploring, I came across another utility in Linux named eTool, which we can help to extract many meaningful or internals of the network interface and also its open source. So we can add new features to it. Ever since I started working on eTool, it was fascinating me and I tried in improving the tool. This is the result of my work and experience with this tool in the past couple of years. In this talk, I would explain what all issues I encountered while working on developing and validation of Ethernet IP. And I have taken the example of design where Ethernet QOS IP and automotive SOC from Synopsys in this presentation. So how this IP is helping to resolve most of these issues and how we utilized eTool to explore and get the best out of Ethernet IP. Okay, let's begin. On behalf of my co-author Ravi Patel, I welcome you all to the presentation on eTool at Diagnostic Approach for Network Issues in Linux. Let me quickly walk you through the agenda of this presentation. First, we will see about introduction to Ethernet. We will see about standard Ethernet and we will see about automotive Ethernet. Then we will see the need of debugging and performance tuning of a network interface. What are all the issues we are commonly facing and what are all the tools available to debug those issues. Then we will see about MAC management counters, what are MAC management counters and how they are beneficial in debugging any issue. Then we will see why we have chosen eTool to implement this feature, what are the benefits of implementing this feature in eTool. Then we will see our work which will contains the implementation we have done in the user space and in the kernel space. Then I will explain you one real use case which we encountered while working in Ethernet. Then that I will show two use cases which will show the result of our work. Then finally I will conclude with the future scope. Okay, now let's have a look on Ethernet, standard Ethernet. We all know that Ethernet is a set of technologies and protocols that is primarily used in LANs. It is it provides a service up to data link layer. At the data link layer Ethernet divides the data stream received from the upper layers and encapsulates into frames before passing them on to the physical layer. Ethernet has gone through four generations like standard Ethernet, fast Ethernet, Gigabit Ethernet and 10 Gigabit Ethernet. Standard Ethernet, it can transmit data at the rate up to 10 Mbps speed and fast Ethernet have a speed limit to 100 Mbps. The 100 base TX standard has become the most popular due to its close compatibility with the 10-dacity Ethernet standard. The Gigabit Ethernet works at 1 Gbps speed which is 1000 base T. The most important difference between Gigabit Ethernet and the fast Ethernet include the additional support of full duplex operation in the MAC layer and at the data rates. 10 Gigabit Ethernet is the fastest and the most recent Ethernet standards. It is 10 times faster than the Gigabit Ethernet and it is entirely based on the use of optical fiber connections. Okay, now how automotive Ethernet differs from this. Automotive Ethernet is a subcategory of Ethernet as specified in IEEE 802.3. It operates over a single differential pair of wires and it is specifically designed for the low-radiated emissions and immunity requirements of the automotive industry. The distance which these standards operates is also much shorter than other Ethernet standards given the size of the vehicle they are intended to be used in. As we can see in this slide automotive Ethernet have crossed three generations from year 2010 to 2020. In 2010 it started with the use of Ethernet in cars for diagnostics and firmware updates. It used 100 base DX standard though this standard does not meet automotive requirements but this interface is only used for diagnostics while the car is in service location and exemption was made to allow its use. In 2015 Ethernet is used for camera driver assist and video infotainment connections. In this model Ethernet is used in point links and is noted being used as a shared medium for different interfaces. Currently in the new model switched 1g automotive Ethernet will interconnect all the domains in the car meaning that Ethernet will be a shared medium. The new anatomy not only helps to reduce cost and weight but also makes it much easier for the different systems in the car to cooperate. Nowadays people expect car to be like their extensions of their home where they can access all the devices they use at home like apps, social media and real time information many more. So automotive Ethernet will be the solution for all these requirements. That's a pretty much about automotive Ethernet. It's time to think about why what is the need of debugging and tuning of the network and what are all the tools available for this purpose. As we are working in Linux probably we must have faced various network problems. Being working with a high speed Ethernet IP the sheer volume of traffic on a super busy network makes it very tricky to diagnose any issues. Whenever an issue occurred we have to obtain all the required information using all needed tools to find out at what level of the network stack contains the bug. For example we can use if config or e-tool to get information about network interface which includes status of the link, speed of load parameters and interrupts occurred and number of packet got dropped during data transmission and so on. We can use IPERF to check the bandwidth availability and to validate whether the whole target is accessible or not we can use ping, trace route or dig with diagnose everything DNS. Another one popular command instead which we used to get all information about TCP connections. Hence being a software people it is very important to know about these tools as if maybe in real time if a product is out in the market maybe we don't have permission to connect JTAG or something to debug any issue occurred at that time all these software tools will help us to debug any issue. Okay let's see about MAC management counters. MAC management counter or MMC counter or the extension of the registered address space of the CSR module. It contains various registers to maintain the statistics of various received and transmitted packets. They also contain control registers for controlling the actions of registers like there are two status registers containing the interrupt generator during transmit and receive and interrupt enable registers for transmit and receive. The MMC counters are free running. There is no separate enable for the counters to start. If a particular MMC register or MMC counter is present in the RTL, it starts counting when the corresponding packet is received or transmitted. The width of each MMC counter is 32 bit by default. So you can individually change the width to 16 bit for a selected MMC counter during configuration. These MMC registers are very useful when it comes for debugging and monitoring different types of packets which is essential in automotive application. Okay, so though many tools are available for network tuning and debugging, why we are focusing on each tool? Each tool is very versatile. It can be used for various scenarios like configuring features such as TCP segmentation of loading, DXR extension of loading and etc. It is important to know about network tuning as we are dealing with high speed NICs like Gigabit Ethernet. As I said before, each tool is an open source utility and can be enhanced as per the requirements of the new hardware features. Each tool isn't concerned with the IP address of VLANs and subnets. Instead, each tool lets us to manage then configure the software drivers and the hardware settings that control the network interfaces. There are so many parameters that can be tuned by each tool. I have listed some of the few basic parameters like we can select the port. We can check and change the transmit and receive mode of operation, either full duplex or half duplex. We can change the offload parameters. We can check and change the speed of transmission. We can control the auto negotiation of the network speed and mode of operation. We can change the physical address and it can be used to set VACON LAN option. Another important feature of each tool is getting the statistics of the network interface. We can get more detailed information of network interface like number of packets transmitted and received and count of various error packets. There is a reason for highlighting this because we have implemented a new feature in each tool, which is like an extension for the statistics showed currently in each tool. We will see about that in detail in further slides. Okay, we can install each tool in most of the Linux distributions. First of all, as we have to do is to update the APT package. Then we can use sudo apt install each tool life and via option to install each tool in Ubuntu systems. And if we are using Fedora or CentOS system means we can use sudo dnf install python 3 each tool command to install each tool. Let's have a look on each tool architecture. Each tool is a command line utility that resides in the Linux user space. It provides consistent access to networking commands directly through bash. It runs directly from and integrates with bash while being interoperable with the regular way of accessing underlying configuration files and automation. As we see in this slide, each tool uses I octal interface to communicate with the kernel. Once the each tool I octal is called, it will check for the relevant each tool of sender respective driver and do the operation accordingly. Okay, so how does each tool is communicating with the kernel? In each tool interface, all informations are passed in the form of request and replace as defined as macros in each tool.h uapi header. The each tool command enables you to query or control the network driver and the hardware settings. As you can see in this slide, each tool request or divided into three categories like get set an action get request or the or sent by the user space application to retrieve some information. They do not contain any message specific attributes. They do not contain any message specific attributes kernel replies with the corresponding information by passing a reply message. If the data has to be modified, that can be done by passing set request with the corresponding attributes, which we have to modify replies to most of the set command will be error code. And if the kernel provides any additional data, it is sent in the form of reply. Most of the cases that command needs pseudo permission to apply the changes. The most commonly used get and set category request action is used in case if the user wants to perform any specific action. So as you see in this slide, for example, if if the get request has been sent from user space to kernel space, like if the user want to get the link info, the message will be sent like each tool message link info get and the response will be like each tool message link info get reply. If the user want to set any property in link info, means the message will be like each tool message link info set. This will be passed from user space to kernel space and the kernel replies with the appropriate notification reply message. And if the user wants to perform any action like here in this case, if the user wants to perform any cable test, the message is passed from the user space to kernel space is each tool message cable test act. And the kernel replies with the appropriate notification message. Okay, let's see how the IOP communication is happening in each tool. The flowchart shows how the user space command each tool ends up invoking the dev each tool on the kernel site. The interface between the user space and the functions is the IOP system call. As we see in this flowchart, each tool uses SAOC each tool IOP to communicate with the kernel. When an IOP call occurs, do IOP function gets invoked in the Linux kernel in which dev IOP function call is happening. Inside dev IOP function, all the IOP calls defined in the Linux kernel is added in a switch statement. So that when SAOC each tool gets called, that particular switch case will get executed. Dev each tool function will get called. All the each tool routines are added in the dev each tool function. Hence, depending on the function called particular each tool routine will get executed. Here we go. We will see what modifications we have done in each tool user space side to get the MMC registered details. We have declared a new option in each tool to get the MMC information like with a flag called showMMC and a function called doGMMC stats. This doGMMC stats function take command as an argument and will call another function called doGMMC. This inside this doGMMC function currently we have used getStringSet function which will do an IOP call to get the MMC registered information and the respective size of the statistic so that we can allocate enough memory. Now we will see the kernel side implementation what we have done to get the MMC registered details. As I said before, SAOC each tool will get called and along with the command of getStringSet. So inside dev it tool function it will check for the command. Since we have passed getStringSet function each tool getStringS function will get called. Inside each tool getStringS function we have added a separate case to call estimate getMMC stat function. This estimate getMMC stat function is defined inside estimateEatTool.c file. In the estimateEatTool.c file, first we have declared a separate structure for the MMC registers which will contain the bitwise information. This structure contains bit index and the bit name. Like this we will be adding a structure for all the registers with the bitwise information. And we have added a small function to convert the decimal value of the register into a 32-bit binary value. In the estimate getMMCStats function, first thing we are doing is read the register value. So first we will read the register MMC registered value. Then we will be passing it to the decimal to bin function to convert into a 32-bit binary value. Then we are checking for each bit, whether it is set or not. And according to that, we are passing the bit name and if it is set, it is we are making us enabled. And if it is not, we are making us disabled and we are copying all these strings into a common buffer. Finally, this buffer is passed back to the user space. So we saw our implementation what we have done in the kernel space and the user space to get the MMC registered details. What made us to implement this? While working in the Ethernet IP, one time we faced the issue like our if config shows these many errors on the TX side. But the pink flood test with a different packet size shows there is no packet drop, but the error status was increasing when the packet sizes more than 100 bytes. And we are having one more issue like in the MTL, we have set the disabled transmit status field. This caused an issue in updating the status field in the TX descriptor. So we are unable to find what error occurred. Does it mean the software won't be able to find what error occurred? It was then while exploring, we got to know about MMC counters. MMC counters have information about the interrupt status, interrupt mask and various other registers to show how many packets got dropped because of that particular error. But the current eased tool implementation does not support the support to show the registered details of the MMC control register or MMC interrupt status register interrupt mask register. We have added support to get the value and the bitwise information of the MMC. We have added extra support to show the register value and the bitwise information of the interrupt status register and MMC control register and interrupt mask register. So using this, we were able to find that underflow interrupt have occurred and we are able to find number of packets were dropped due to underflow error. Then this helps us to resolve the issue. As we saw our implementation, now we will see our first use case. We have taken MMC control register in DWC equest registers is the very first and the basic MMC register, which have bitwise information that this very useful for debug purpose. This MMC control register value is not displayed in the current tool implementation for displaying MMC registers. We have added additional support for the MMC control register so that user gets more details information about the critical register in the user space itself. As you can see in this slide, this is how the information looks like when we give a each tool command. We have given show MMC flag and the device name as E0. You can see it displayed the first the value of the MMC control register and shown the each bitwise information like particular bit is enabled or not. Next next use case we have taken is for MMC RX IPC interrupt mask. The current implementation of the E2 shows this value of the register as a big number as shown in this slide. It is very difficult for the user to decode and interpret the bitwise information. We have added additional support to get all bitwise information of this register. As you can see, we can we first we will get the value of the register then each bitwise information like a particular bit is enabled or disabled. So we saw our implementation, output and what made us to develop this. Now we will see our final conclusion and future scope for this work. We are able to enhance E2 user space utility to capture some of the critical debug registers of design where equals controller and present a readable content instead of just dumping the register values. We modified the design where EQS driver in the kernel by adding a required E2 hops hooks to get all the MMC register value and the bitwise information of MMC control register interrupt status register interrupt mask registers as defined in EQS IP. This implementation is helping us to get all the information via E2 command line. The user can simply use this tool to tool for their self diagnosis. Future scope involves implementing this functionality in E2 for other MMC registers and upstream in the source code to the Linux mainline community. Please feel free to ask any doubt regarding my session. I thank you all for joining the session and I thank whole Linux Foundation committee members for giving us this opportunity to showcase our work. I hope our work will help the users to reduce some of their burden in debugging this in some of the issues which needs MMC register details. Once again, I thank you all for attending this session. Have a great day.


From the heart of the jungle comes a savage cry of victory. This is Tarzan, Lord of the Jungle. From the black core of dark Africa, land of enchantment, mystery and violence comes one of the most colorful figures of all time, transcribed from the immortal pen of Edgar Rice Burroughs. Tarzan, the bronzed white sun of the jungle. And now in the very words of Mr. Burroughs, the story of the end of the world. The equatorial sun was strong, and yet as it filtered down through the heavy foliage of the jungle, it was not uncomfortably hot. All was well with the world, and Tarzan was happy.  Yes, it was a good life. Tarzan smiled down at Togo.  Are you sure you will not be lonesome? Togo not a day, Lo. Grow man now. Of course, of course. Only even grown men sometimes miss their home when they're away. Togo not miss home? We'll be too busy hunting elephants and panthers and wild boars. Well, we will have to hunt for food, and you will have a chance to practice your skill with your bow and arrow. But, Togo, you must remember this time that animals are to be killed only for food or in self-protection. Togo, remember? While Tarzan away, Togo protect tribe against Bulgani the gorilla and Numa the lion. Togo very brave and strong. Well, of course you are. But Tarzan is tired, so if Togo doesn't mind, we will have a short rest now. Well, if Tarzan tired... Oh, no, no, I wouldn't sit there, Togo. You're about to sit on Kota, the tortoise. Where? Kota? Is Kota dangerous? Tarzan... Oh, no, no, of course. Kota will not harm you. He has heard of Togo's reputation for strength and bravery, and even now he runs from you. You're not move very fast. Well, it's running for him. No, he won't hurt you, Togo, but I think that moss-covered log would make a better seed. Taking a rest wasn't a bad idea. Tarzan, you look funny. What is wrong? I thought I heard... Yes, it is the cry of Tantor, the elephant. He's angry. Tantor angry? This is strange. I catch the scent of the termungani, too. And yet an elephant seldom attacks a man. I must hurry to that spot. Someone is in grave trouble. Don't leave Togo. Of course I won't. Jump on my back and hold your arms around my neck as tightly as you can. And remember, we must both be brave, whatever happens.  When obstacles loomed in his path, he grasped the hanging vine and hurtled over them. Now the scent of the elephant and of termungani were closed. Breaking into a small clearing, Tarzan could see the man kneeling upon the ground. His hands folded in prayer, and rushing toward the helpless target was Tantor, gone mad. But at the sound of Tarzan's savage cry, the call of the bull-ape, the elephant swerved in its path and wheeled to face a new enemy. Now a charge straight at Tarzan and Togo. But just as it was about to strike, Tarzan grasped an overhanging vine and swung out of the path of the rampaging elephant. The termungani is Mr. Martin, the missionary. Thank thee, Father, for sending Tarzan to save me. My prayers were answered, and I may yet be of service. Mr. Martin, take care of the boy. I must capture the rogue elephant before he kills anybody. Oh, no, let the animal go. What?  There is a far greater threat to life on the rampage. I have traveled many days to find you, Tarzan. Only you can save the people of the jungle from this new threat. My instincts tell me to follow that elephant, to kill him before he kills others. Believe me, the enemy of which I speak is greater. Even now the talking drums of the jungle spread the message of horror, and soon there will be ought in the jungle safe panic. In just a moment we shall return to our story of Tarzan. They formed a strange picture, those three in the tiny clearing of the jungle, a small native boy with round, wandering eyes, by his side Tarzan, bronzed and powerful, and before them the slender, aesthetic-looking missionary, the Reverend Thaddeus Martin. As you know, Tarzan, I've devoted my life to missionary work the last eight years among the people of the Negalia tribe. You have done much good work, Mr. Martin. But you've not told me of the danger that you say threatens the whole jungle. We'll have patience just a moment, Tarzan. I'm getting to that. Oh, I'm sorry. Torgo, sorry too. What have you done to be sorry for, Torgo? Torgo did nothing, but if Tarzan's sorry, Torgo's sorry. All right, all right. Now we're both sorry. We will both be patient while Mr. Martin tells us his story. Well, as I was saying, I've spent eight years among the Negalias. To all outward appearances they had become good Christians in their way. But a few weeks ago they came to me with a fantastic story that we're soon to witness, The End of the World. The End of the World? They've deserted my small church and they've returned to their pagan gods. Each night the jungle throbs with their frenzied drums and their maniacal dances. The Negalias' old dances? A kind you will never see, I hope. It's spreading, Tarzan. Their talking drums have broadcast the fear to other tribes. And by the time the full moon has come, my work of eight years has come. My work of eight years will be wiped out and my people will return to complete savagery. But what can I do? Well, perhaps you can find out what and who is behind all of this. It's not the witch doctors, nor the snake men, nor the porcupine men. That much I know. But who else could originate such an idea, The End of the World? Oh, I don't know. I've questioned everyone in the tribe. They are too frightened to tell me anything. Mr. Martin, I will come with you. I will see what I can find out. But first I must return Torgot to his people. The delay of a few days might mean The End of the World in a different sense, but violence may break out at any moment. If End of World comes, Tarzan, it's not matter whether Torgot home in Land of Punyas or in Land of the Negalias. Every minute counts. Well, if this panic is spreading as rapidly as you say, Torgot will be no safer at home than with me. If you are ready, Mr. Martin, we will start for the Land of the Negalias. All that day, Tarzan and Torgot and the Reverend Martin marched through the jungle. Several times Tarzan caught the scent of the rogue elephant, but there were more important things at hand. That night they rested briefly, and by noon of the next day they approached the crawl of the Negalias. To all appearances, it was indeed the village of a people awaiting death. In the Shamba, weeds choked out the young corn. Rodents overran the narrow paths between the cone-shaped huts, and the huts themselves were those of a people too sick in mind or body to care for their homes or their children. This is the site that meets the eye by day, Tarzan. At night there is abandon. You still have a room for sleeping attached to your small church? Yes, of course. It's but a short walk from the village. And if you will take Torgot there, I prefer to talk to the people along. Torgot, I want to leave Tarzan. I may need your help later, Torgot. It would be better if you were well rested. All right, then. Come, Torgot.  Peace with you, Tarzan. What white men want? Negalias not want white men. Wait, not wait. End well, come soon. All white men go to Lubuga, Vassia, Vassa. I have no intention of going to the inferno in the near future. The end of the world is not at hand. The sign's plain. End world, come at full moon. Your breath smells of much congala and your words are those of a man congala has robbed of reason. Where is your chief? Me, Sir Barley, chief of Negalias. What do you want? I am Tarzan, lord of the jungle. I come because you spread fear through the land. This false story of the end of the world. Is true.  Tarzan is your friend. He takes the armlet from his arm and gives it to you as a sign of friendship. Sir Barley, not take. There's no ornament to give in place. You speak in truth. You wear no ornament. There is hardly an armlet or a nose ring or anklet among your men. What is the reason for this? Tarzan asked too many questions. Better end his world now. Wild elephants run for life. Tarzan stalking Tarzan's party ripped through the village with savage fury. Native huts and scattered debris fill the air. The natives ran in all directions, escaping through some fantastic miracles. And high in a tree, Tarzan watched the sea. He could not spare time even now to hunt the deranged pachyderm. For his eyes must remain upon Ugar, a one native in the village who still wore ornaments of metal. That night, when the tribe had dropped from exhaustion after hours of reveling, he saw Ugar walk from the crawl. Tarzan followed him and saw him enter a crude cabin several miles away. The lord of the jungle crept to a battered window and peered in. The native quake as he stood before a white man. Well, it's about time you got here, Ugar. Expect me to stay up all night? Came soon I could. What did you bring this time? Here. Many rings, bracelets.  Why didn't you bring some good stuff? Not much left in village. Already you have most. There's still plenty. You see this coin? In the deep. Now watch. I wave my hands and it's gone. So. Where'd it go? The great Zaker has caused it to be no more. I am the greatest magician alive and if you don't bring me some real stuff tomorrow night, I'll make you disappear in exactly the same way. Tomorrow night, Ugar, try to bring more. What you see in crystal ball today, great Zaker?  No, no, don't send Ugar to land of demons. Our people send you all of rings and other ornaments. Ah, you'd better. You see this handkerchief? The D. What color is it? God. Red. Now I pass it through my hands. Green. So can I change the fate of your people. When the full moon comes, you will go to the land of your ancestors or to the land of the demons. It's entirely up to me.  I'm paying you for them. How much you pay for what Ugar bring tonight? You've told no one outside of your tribe about me. No, I swear. Good. I shall pay you ten bottles of kangala. Here, take them. Let your chief distribute it. Go now. I go. Now that's another tidy sum of... Don't turn around or you'll feel the steel of my knife. Who are you? What do you want? I am Tarzan. Now that I see your game, I want a great deal. Yeah, sure. We can make a deal. The same deal you're giving the natives, trading cheap whiskey for their jewelry? They get their monies' worth.  I should say that the native junk, much of it made of pure gold and set with precious and semi-precious stones, should be worth a good deal of money in distant ports. Okay, so I found a good thing. Look, let me turn around and talk to you. I don't talk so well when I've got my back to a man. You can turn around, but my knife remains unsheathed. Sure, sure. Well, so you're Tarzan. I've heard a lot about you. You will hear more unless you go to the native village with me and admit that your stories of the end of the world are nothing more than part of your magician's act. No, you're wrong. My prediction is true. I sought my crystal ball. I don't know where your crystal ball came from, but I know the amateurish tricks you showed the simple native tonight can be bought in any bazaar in the country for a few coppers. You speak hastily. I speak with knowledge. Such tricks are advertised in magazines in many countries and can be ordered by mail. Well, a few of them perhaps, but I also have many important feats of magic. Let me show you this one here. Stay at arm's length from me. Sure, sure. Observe, my friend Tarzan, a great feat of magic, a mystical accomplishment of Legend of May. You see this glass of water? Yes, I see it. I pour into it this small amount of white powder. So... And now the water is red. I suppose you're going to tell me that you have turned water into wine. I think that's the usual speech that goes with that trick. Oh, that's not the end of the trick. Watch carefully. Keep your eyes on the glass. See? Now I take the glass in my hand and I... Oh, my eyes! My eyes! I've been blinded! You see, Tarzan, you were wrong all along. I have some fine tricks and you were wrong, too, about my prediction of the end of the world. It will come soon... for you. In just a moment, we shall return to our story. In the small shack of Zaker the Great, Tarzan sat on the edge of a soiled cot, his head hanging and held between his great hands. His legs were shackled to a thick iron ring in the wall. But this did not bother him nearly so much as the horror of blindness. The loss of one's eyesight is a tragedy to anyone, but it seemed an even more overwhelming catastrophe to one whose life was spent in the wiles of the jungle. For once, Tarzan was beaten. You want something to eat or not? No. No, nothing to eat. You really ought to, you know. I fixed a mighty tasty dinner. A bit late for dining, perhaps, but I've had a lot of things to get ready for tonight. This is the night of the full moon? Oh, yes. Yes, there's a nice full moon. Too bad you can't see it. Say, this is good.  Got a couple of full bottles here yet? I want nothing. Seeing this is your last night on earth, you're sort of upsetting a tradition.   Why are you dragging out the torture? I've got a little plan for you. You're going to help me get rich. I don't know what fantastic plan you have, but what good will riches do you if, as you claim, this is the end of the world? Oh, you didn't really go for that stuff. No, no. The natives did, though, but they're stubborn. They still haven't told me where they got the gold they make those ornaments of. So, once again, gold is the source of trouble. You should see some of the nuggets they've used to make necrosis. A few hours in the place where they get them and I'll be fixed for life. By the morning, the natives will know that you have told them nothing but lies, for the world will still be here. But I won't. By the time this term is abated, I'll be well on my way to Mombasa. A storm? I hear no storm. Zakir the Great, in addition to his magic, is also a student of science. Between my almanac and my barometer, I've made out that tonight there will be a great equatorial storm. But what has that to do with... I've built for this for weeks now. The Nagalya's expect the end of the world. When the storm breaks, they'll be sure it's come. Then I will enter their village with proof that I, Zakir the Great and powerful overall, then they will tell me where the mine is. They will do anything I command. And the proof you speak of? I will be leading the lord of the jungle on a chain. And they will find that the man no one else has been able to conquer, has been transformed by Zakir the Great into a weak, crawling, blind creature. Torgo and the Reverend Martin had come to the village of the Nagalyas searching in vain for some trace of Tarzan. They could find nothing. Nor could they extract any sort of an answer from the drunken, frenzied savages who danced about the Tribals. They could find nothing. Nor could they extract any sort of an answer from the drunken, frenzied savages who danced about the Tribal Files. It's no use, Torgo. They won't listen to us. Must find Tarzan. My resources are the name. I hate to admit failure, but... Torgo not at the end of his horses. Torgo has strong teeth. Torgo, what are you doing? Ow!  What did you do? You bite Ugar's leg. Boy, gag and title-staken center-counsel ring. You're not tying the boy up, Sibali. Take boy away. Sibali, surely you have not forgotten all of the teachings of our Lord? Not Lord of Nagalyas. He not stop end of world. There will be no end of this world tonight. The full moon has come and still is... His end of world. Like Seiko, the great Se, end of world here. Let me get to the boys. Stop, thee boy alone. You not good. Tell us many lies. I'm going to get that boy if it's the last thing I know. May you pick up missionary. Drag him to Hima. Let missionary stay on ground where he is. Look, Seiko the great com. Brings prisoner. His Tarzan. Friends, at the end of the world has come that Seiko has told the truth. His truth. And now I call your attention to this weak crawling creature. His Tarzan. Yes, his Lord of Jungle. No longer is he Lord of anything. I have taken his power in his sight. It is a sign. Seiko the great has power over all of you. He will deliver you to the land of happiness or he will send you to the demons. Don't send you demons. Save us. If I am to save you from the demons, you must have no secrets from me. Where is the gold mine? Another moment and it'll be too late to answer. Now tell white man where gold. I warn you, unless you answer at once, you'll go to your doom like Tarzan. Blind and helpless. No, not blind. I'll take like Tarzan. We tell. Place where we get gold is behind. The lightning and the thunder and the screaming have brought the wild elephant to fever pitch. As he entered the village, threes were smashed as though they were matchsticks. The boma was torn aside by enormous yellow tusks. Native huts were demolished. Parties were picked up in the great leathery trunk and then hurtled through space.  Save for a small native boy who was tied in gag, a man of God who lay unconscious on the ground, a blinded giant, and a man to whom he was chained. And then the miracle happened. The elephant grasped Zaker in his trunk, breaking the chain that held him to Tarzan. Zaker was thrown against a huge rock and then the mad animal charged in the direction of Torgall. Torgall! How's your saying you, Torgall? Tarzan staggered a few steps. His arms outstretched in the gesture of the blind. And then suddenly the hands came down. He grasped his knife and hurtled himself through space. A moment before the elephant reached Torgall, Tarzan catapulted at the animal his knife, rising and falling, stabbing, twisting, piercing, killing. The gag out of mouth just contained. We must see if Mr. Martin is... Oh, there he is, on the ground. Mr. Martin, Reverend Martin! Wake up! Wake up! Torgall, down in this cloth, in a small stream, right there. Hurry! Surely heaven has protected you, Mr. Martin. I see the Prince of Tantor's hooves not an inch from your head. You're closer to this big cloth. Something.  Tarzan? Yes. Yes, it is Tarzan. Tarzan, who must tell you the story of a miracle. Do they come, Tarzan? Yes. Come men of Nagalia. Your friend and teacher wishes to speak to you. I... I think most of them are here now, Tarzan. Those who are left. Chief Sebali? Man of Nagalia? Once again, you have been hurt by the greed and evil of a white man. But just as there are good and bad black men, so there are good and bad white men. The evil Zaker is dead. The good Reverend Martin lives. Tarzan, who is a friend to all in the jungle, begs you to hear him. Speak, missionary. Brothers, for a little while you were misled by a man who showed you tricks. Man made miracles. This morning, as the sun shines down on us, we realize that miracles cannot be made by man. In the darkness and the rain some strange things happened here. A small boy who was bound found great strength and managed to tear a gag from his mouth. A deranged animal ran wild and yet his footsteps were guided so that I might be spared to help you. And a man who was blind was permitted to see again because he was needed. Perhaps men of science might explain this phenomenon. They would use words that I understand no better than you. Words like adrenaline and shock and stimuli. But I think all of us have a much more simple explanation. Reverend Martin, some your words we not understand. But what you mean? We understand much. I do not think you ever really believed the world was to end for if you had, there would have been no need to run from the elephant. Now we believe only what Mr. Martin teaches us many years. If you believe, join me in the prayer I have taught you. Our Father, we charge in heaven, hallowed be thy name, thy kingdom come, thy holy dawn in earth as it is in heaven. We give you those who say our name, for give us all the best as we forgive you our letters and leave us not into temptation but deliver us from evil for thine is the kingdom and the power and the glory for ever. Amen. In just a moment we shall return to tell you about Tarzan. The girl had sung in a low, vibrant voice but it was not the magnetism of her song alone but the terror in her eyes that brought Tarzan to her aid. With enemies in constant pursuit ever threatening to still the lovely voice forever and to kill Tarzan for his interference he manages to delay her inevitable fate as he guides her across a continent in our next story. Tarzan, a transcribed creation of the famous Edgar Rice Burroughs was a white junior prepared for radio by Bud Lesser with original music by Albert Lesser. This is a Commodore production.


Sorry, let me make one comment about the web design problems or the polar area problems like keeping these emails. It's important, it's critical that you think about what the integral represents so that if you're finding, you have some function in polar coordinates that looks like this and I want to find this area, I have to remember that when I do a polar integral, I'm adding up things that look like this and so you really have to think about what the integral represents geometrically. So for example, there's a problem where you're supposed to find the area between two circles that look like this, it is if you just write, integrate from where the two things intersect to where the two things intersect and take the squares and subtract them and zero because really what you're finding is you integrate from here to here and you get this area, minus this area, well that is equal so that you can do it. So this problem which probably a lot of you have not done so you don't have a clue what I'm talking about, if you solve, so this is like r equals cosine theta, r equals sine theta so I'm not telling you how to do it, I'm telling you what not to do, if you solve where these two things equal, then you get theta is pi over 4 where theta is pi pi over 4 and if you just do something silly like write the integral from pi over 4 to pi pi over 4, 1 half cosine squared theta minus sine squared theta, the answer is zero, that's not the answer to this question and that's the casualty that's represent is the area, cosine squared, this area minus this area, well it's zero, think about what's going on. Okay, so let's move back to infinite series or infinite sums, so as I said before, I will try to say the word sum instead of series, the book does not use the word sum, it uses the word series so if you, the reason I prefer to use the word sum is because it means adding, series means something slightly different in English than it means in math, so anyway. Last time, we saw that if we have something like this where r is a number and so it's value less than 1, it's a geometric series, or geometric gas, geometric series so this is 1 plus r, r squared, r cubed and on forever so what we saw last time is that this adds up to that so that means that if we see something that looks like obviously this 8 thirds plus 16 9ths minus 32 27ths plus what's next? 64, 81st minus blah blah blah, what would you think? How would I do this? So I'm not using the clicker so I can't ask this if it was a question, but that doesn't mean you get off from thinking, so I'm going to tell me how to move this problem. Okay, so what do you see that should happen? What's the numerator of the problem? 2 to the n? Well, but it's not every 2 to the n, you're going to need your clothes, that's not 2 to the 2 n. Yeah, okay, sure, this is not what this equals, we're closed. There's a negative, there's a negative, so is it this? I'm going to start with 0 to the 1. Well, of course it matters, but so let's just try and figure out what it is and then we'll turn it into something that looks like this, so this is not right, what's wrong with this, okay, or I can just, all right, so if you're saying do that, and what is n, sorry, 0, so the n is 0, we have positive 2, the n is 1, we have minus 3, so that seems to work, so okay, and this doesn't look like that, but how can we make this look like that, or do we have any hope of making this look like that? So what's the problem with this, why does this look like that, right? So how can I turn it into being the same power? Yeah? You move over here, what's wrong with you? Why the place that's in there? No, okay, fine, so go ahead. You can make 4 into minus 3 times n, you can split the power. Split the power, right, so that's what I mean. So write this as, so that's around this power of 2, so that I have minus 2 to the 2 times, minus 2 to the n over 3 to the n, all right, that's what you were saying, I hope? Okay, so then that is, well, minus 2 to the n is 4, and it's 4 times every single term, so I can just factor the 4 out, just by playing around with this stuff, I can factor each term out. Now if you're more comfortable working with the initial thing, you could factor the 4 out of everything up there first. The first term is not 1, you factor the 4 out of everything, so if you refer, you could also write this as 4 times 1, minus 2 thirds, plus 4 ninths, minus 8, 27ths, plus 16 eighths, minus 1, right, just factoring the 4 out of this, I get that. It's the same thing, so now this is easy, so what is this sum? The formula's on the board, it's just a matter of noticing and plugging in, how many of you are planning to major in engineering? Okay, so you should be really good at looking at formula, you know, and manipulating them and turning them into specific applications, engineers do that a lot. So what is this sum? What do you do with engineers? Come on, 3 fifths, how do you do that? Yeah, okay, so that's 3 fifths, why is that wrong? Because it's times 4, so it's 12 fifths. It's close, the bridge fell down, but okay. It's close, it's both nice, it's what it fell down for. Okay, so these kinds of things, they're not, you know, so horrible, so it consists of a lot of looking at it, manipulating it, turning it into something that you know, going from there. Notice that we can also do something like, if I had, maybe, what if it doesn't start at 0? Suppose I have something like, let's just do the same one, n equals 4 to infinity of minus 2 thirds, well, let's do this, one fifth to the n. What can I do with this? So this starts at n equals 4, so this begins, I don't even know, 1 over 5 to the 4 plus 1 over 5 to the 5 plus, blah, blah, blah, blah. Okay, you could do that. So this is the same as n equals 0 to infinity of 1 over 5 to the n minus, this song, well, let me just write it, minus the first three terms, four terms, 1 plus a fifth plus 25 plus 125th, so you could do that, so that's, I guess I'll just, I guess I'll write it over here, is the same as 1 fifth to the n, so that's 1 over 1 minus a fifth, which is 5 fourths minus some junk that I don't want to add, so it's 5 fourths minus whatever that junk is. I do that right? Four fifths, 5 fourths, yeah, so it's that. Another way you could do it, which is a little easier for me to actually do the arithmetic, so what's another way? A here is the first term, so you plug in 4 for n, 1 fifth to the n. Wait, say that again? A is the first term, so you're very certain there's 4, so you plug in n to 4, 1 fifth to the 4 to the first term, and then that over 1 minus a. Okay, yeah, so in other words, factor out the first term. Right, so this is the same thing as 1 over 5 to the 4th times 1 plus a fifth plus fifth squared plus blah, blah, blah. So this is 1 over 5 to the 4th, now this is the standard geometric sum, this is 1 over 5 to the 4th times 1 over 1 minus a fifth, which is whatever the heck it is, 1 over 5 to the 4th times 5 fourths is whatever that is, 1 over 125 times a.  Okay, so that's one thing we did last time. Another thing we did last time was the harmonic series. This is a very important fact, like it should add up to something in fact it gets as large as you like. The way I show that is by grouping the terms and noticing that I can make them, I'm computing the partial sum and then noticing that I can make them as large as I want, they just get large slowly. I'll come back to this later. I'll prove it a different way in a minute, several minutes. So let's look at some other kinds of a series. So let's think about, when we're thinking of, I mean, let's look at an example like 3 plus 8. You can start anywhere, how about 0? Should this converge or diverge? And how can we tell? So he thinks it should converge. Yes, right. So it actually diverges. Why is it diverged? So for n really big, this looks like, in other words, so what this squishy statement for n really big is I just move over here, the limit is n goes to infinity of n squared over 5 n squared plus 8 is a fifth. And so when I start writing it first level, so when n is 0, that's easy, that's 0. And when n is 1, that's 13. And then there's a lot of stuff that I want to, and then I wait until about the millionth term and it's something, there's this 0.2.  And there's infinitely many of these things. You add up a lot of 0.2 as you get a very big number. So more generally, this tells us that if, well in this example, this is not a proof, but it's easy to prove, if it doesn't matter what we sum over, if we have an infinite sum like this converges, then for sure it has to be 0. Or say in another way, not 0, not 0, let's say there's a limit. If the bit of the term is not 0, then it's important to realize that it doesn't mean if the limit is 0, then for sure it converges. I can only say if it's not 0, then it diverges. Is this clear? Pretty early because she already knew it. Is anyone confused about this? Okay. So what we're saying is if the terms don't get small, when you add a lot of them, you get it big. If I start adding a lot of things, each of which is not really honey, then I get a lot of stuff. The only way that I can add an infinite amount of things together and have something finite is if they get small. That's what this says. Yeah. I don't care about the 8 when n is 1,000. So when n is 1,000, if I have 1,000 squared over 5,000 squared over 5 times 1,000 squared plus a little about 2,000, a lot of that's good. I mean it's 0.2, it's actually 0.19999999 out of the blue. Very close to a 5th. So, you know, when we're talking about 50 cents in the federal budget, nobody knows.  This is a penny. This is $10. Or $100. Okay, so one fact here is that when we add infinitely many things together ultimately those things have to be tiny and they have to be as tiny as they want so we want it to add up to something. Let me turn to another example. Suppose I have something like this. So I mean part of what I'm doing right now is I'm just putting a bunch of examples up and I'm messing around with them and showing that we can do something. We'll turn to developing some theory precinct but suppose I have something like this. So this is, so when n is one I get a half. When n is two I get a fifth. When n is three, six, one over four plus two, you're right, a sixth. When n is three I get nine plus three is twelve. When n is four I get sixteen plus four is twenty. And so anything I can do here. Notice that it has a chance of converging because the limit as n goes to infinity of this is zero. So it has a chance. Maybe it converges, maybe it doesn't. How could we see that it converges? In fact I can see what it has to do in the hand of the one. Yeah. Well if I replaced everything with one sixth then I would get an infinite number. So if I replace everything with one sixth I would have one half plus as many sixths as I can. Well then I would show what a diver is. Is that what you're saying? So you're saying replace these with the sixth. That's pretty soon to fail. You can try it. It's not going to work. So I can tell you at the beginning that this converges. It is obvious that it does converge. It's obvious that you don't have to do this stuff. It's not obvious. It shouldn't be obvious yet except for your refugees from 127. Okay. So it's not obvious even what to do here. So a trick is we can use partial fractions. Remember those? So maybe we can do something by rewriting these in a clever way. So if we have that, so that means that A times N plus one plus B times N equals one. Right? Using partial fractions. B is for integration. And so this means that we have A plus B is zero because I have an N here and N here and no N here. And A is one. So B, right? A is one. So B is minus one. So that means one over N squared plus N is the same thing as one over N minus one over N plus one. And so now, I guess I'll go back over here. So using that, write this this way. So those are the same by algebra. So now if we start writing out some terms, the first term is one minus. And then the next term is a half minus a third. And then the next term is a third minus a quarter. And then the next term is a quarter minus a fifth. And so on. And notice what's happening. I take a half, I throw a little bit away. I take one, I throw a little bit away.  Then I add that amount back and I throw away a different amount. And so on. So this is one. And if I group these together, I get nothing. And if I group these together, I get nothing. And if I group these together, I get nothing. So this is one. This is called a telescoping series. Or maybe telescopic, I don't know. And it should be like, you know, in the pirate movies when they take out their things. And then they close it up again. So this expanded little big thing, which you could just collapse. So this trick or technique works sometimes, but not all that often. Once in a while, it's going to be a handy thing. Now usually, usually actually finding the sum is not so important. Sometimes it is. Finding it exactly is not so important. What's important is knowing whether it does have one or not. So a lot of our focus won't necessarily be on when these things add up, what do they add up to? Because I can use a computer to do that. If I want to know what this adds up to, I can find it to a thousand decimal places by just running a computer for a little while. But if it doesn't add up, I'll get an nonsense answer. So the real question, a lot of the time, is not what is the sum, but is there one? So almost more important than what the sum is is should I even bother to look? Serious. Less important. Less important. It's also harder. Well, it's harder to get exact and not so hard to approximate. So this is a very important question. So a lot of times we'll just focus on does this converge? Yes? No. And another thing that's important is to be able to convince somebody that they're not just flipping a coin and saying, yeah, it converges. So you have to be able to justify your answer. So, see, this is already an important thing. If the limit of the nth term is not zero, then it doesn't converge. We're done. We already have one of these kinds of things. And let's work on developing a couple others. So, and let's work on it with a specific example. The quantum series, sum of 1 over n, does not converge. Let's look at what I started with. So it squares plus a quarter plus a ninth plus 16 and so on. Now, does this converge? It has a hope. It has a hope of converging because certainly the term is zero. It also has a hope of converging because it looks kind of like this one. Looks a lot like 1 over n squared plus n. So, and in fact, just using this path, that one is, well, it's not quite good enough. Right? So this looks like whatever this means, and I'll make this precise in a few minutes. Well, no, probably next week.  We know this one is one. So we would expect that probably this converges. But expecting that probably it converges is not good enough. We have to know that it converges. But how do we maybe know that this converges? Notice that we can't say, well, every term here is less than one of these because actually these are small. This guy is still here. The first term there is one. The first term here is a half. The second term there is a quarter. This term is six. So I can't say, well, this one's bigger than that one's with flushes with n because actually that one's bigger than this one. But they're kind of related. The trick is directly comparing isn't going to work. So we have to try something else. So what do we do a lot in this class? What have we been doing all semester? Integrating. Maybe integrals will have something useful to tell us. So let's draw a picture. Well, let me draw a picture. You don't have to draw a picture. I'm going to draw a picture and I'm going to think about what that picture is telling me. Let's take a bunch of rectangles with one, one over n squared for the various values of n.  So here is a rectangle of height from zero to one of height one. And then next to it is a rectangle which I'll put between one and two of height one over two squared. And then next to it I will put a little rectangle of height one-ninth. And then I will put a little rectangle of height one-sixteenth.  But I have a bunch of rectangles that look like this. Why did I do this? I don't know. This is math class and we do weird stuff in math. What do we know about this area? What function is it related to? What have I done? This point, this point, this point, this point, this point, this point, all live on some curve. They live on the curve y equals one over x squared. This area here is certainly less than some integral. What integral? Somebody said it. Yes. It's less than the integral one over x squared from one to infinity. So it's less than, I guess I'm going to write it here, the integral, well you can't see as I write it, the integral from one to infinity of one over x squared dx. But we already dealt with those kinds of things. Well, we did it. Some people got it right. So we know that, I'll put it in quotes, we should know that the integral from one to infinity of one over x squared dx, this is the limit, as m goes to infinity of that integral, which is a number. So this is, why can't I do it? x is a minus two, so that's one over x negative from one to m limit. So that's, so we know for sure it's less than one. So we know then that this adds up and it adds up to something less than one. So now if we want to figure out what it adds up to, you can just take out your computer and start adding and you get a lower bar. Oh yeah, somewhere I screwed up, didn't I? Where did I screw up? Oh, because I did zero, no. What did I do wrong? Oh shoot, it's certainly bigger than one, isn't it?  What did I do wrong? I did wrong. Notice that the integral here, I started at one, so this rectangle wasn't included in the integral. The integral is this integral, but my sum has a one more. The integral only gives me this part. So we'll see that. Oh, it's less than two, thank you. It's close. But it certainly converts. In fact, if we want to get it bigger than something, you can put the boxes on the other side. But a terrible purpose technique, unlike, say, this telescoping something that only works if you're lucky and stuff cancels out for any integral. So let's see what we can do. So notice that this trick, this technique that I used here, didn't really use anything special about the specific series other than all the terms were positive. And I compared what the series is doing to what the integral is doing. So it doesn't really matter what the function looks like as long as all the terms are positive. So let's see if I can write that. So if the terms are always positive, and if also, well, let's see, and also my terms are some function, I'll rewrite this in a nicer way than that. Then if the integral, I want to go infinity and it doesn't really matter where I start. I'll start with one. If f of x dx converges, so does the sum. We can draw this picture. The sum is an approximation of the integral. In the picture, f of x is decreasing than the sum plus the signature. So the sum and the integral are closely related. It goes the other way, too. If the integral does not converge, neither will the sum. So let's look at that. So let me do the one that we know diverges. And then I'll say it again in a nicer way. The one that we know diverges. So we already know the sum of one over n diverges. We already did that by cleverly putting terms together. Let's compare that. We also know that the integral, well, I hope we know that this diverges when you do this integral, you get the log and the limit as x goes to infinity of the log is not zero. It flows up. So we did this before in class, I don't know, a month ago. We saw this diverge. You won't remember that. Let's relate these by picture again, but not the same picture. So here's the graph. y equals one over x. Let's put the points where my scale is off but I don't care. Here are the points. One half, one third, one fourth, and so on. Let's again put little rectangles on there. We don't have to start our little rectangles at zero. Let's just shift them over by one. Put them here. Now I'm putting the rectangles on the other side. So this area is bigger than the integral. But what is this area? Each of these I'm choosing to be one y. This is one plus this area is a half. This area is a third. This area is a fourth. And so on. That's my sum. So by thinking about it this way, we see that the sum of one over n is bigger than the integral of one over x. So let's put this on the other side. The integral pushes it up. And this is different. Now I didn't prove this but you can turn this into like a real mathematical proof. But let me just restate it the same way. This is something called the integral x. We just want to make sure I didn't leave out something. Write it a little more carefully. So f of x is a continuous decreasing function, decreasing function f of x. And we have f of n equals a of n. In other words, that's what our series is made up of. Then if the integral, it doesn't matter where you start from somewhere to infinity from f of x dx converges, so does the sum. If the integral of f of x dx diverges, so does the sum. Well, this is sort of the story that we've seen I see right now. This is the story that we've seen from the beginning. Integrals are just special sums. So if it's infinite, if it's improperly convergent, so does that. If it diverges, so does that. Okay, I'll see you my day. Have a nice weekend.