Hello and welcome to a screencast today about finding the integral of convergence for a power series. Okay, so the power series we're going to look at today is the sum as k goes from 1 to infinity of x minus 1 to the k divided by k plus 2 times 4 to the k, and that 4 to the k is in the denominator, just so you know. Okay, on Geogebra I graphed our different partial sums for this particular power series. k equals 15 is in red, so that's kind of this outer function down here. k equals 25 here is in blue, kind of has the basic same shape to it, but maybe a little bit tighter. And then 50 looks totally different, well kind of totally different. On one end it looks different. It's here in black. So you can kind of see then, so our power series is centered around 1 because we're subtracting a 1 off of that. So then it kind of looks like things start to go apart here at negative 3, and sorry you can't really read that number, but that's negative 3, and then maybe somewhere in here in the 4 to 5 range, things kind of start to shoot off too. So it gives us a good visualization of what's happening. So we can kind of have a gut check maybe to see if our interval of convergence is correct or not. Okay, so here's our power series, and again remember we want to go ahead and use our ratio test. So let me remind us of what that is. So we want to look at the ratio test, which says we've got the limit as k goes to infinity of the absolute value of a sub k plus 1 all over the absolute value of a sub k. And remember this is going to converge when this limit is less than 1, which is obviously what we want. We want the interval of convergence, so that's really what we're looking for here. Okay, so let's go back, oops I guess I shouldn't have gone down too far here. Here is our a sub k, all the stuff here inside our summation. So a sub k plus 1 just means that we're going to add 1 wherever we see a k. So that's going to give me, and then if you've watched a video before with these absolute values, I'm going to go ahead and take some liberties here with them. So I'm going to write the absolute value of x minus 1 to the k. K is always positive, so the absolute value of that's not going to affect anything. Divided by, same thing with our denominator, okay, oops and I forgot to add my 1. Got too busy talking about the absolute values. In our denominator the same thing is going to happen, right? Everything here is positive, so those absolute values are not going to affect anything. So I'm going to go ahead and write k plus, let's see, so k plus 1 plus 2 is going to give me k plus 3, and that's a k, and then we're going to have 4 to the k plus 1, okay? Alright, that's just in our numerator. Then we're going to divide by, I'm going to do something kind of similar in my denominator, so that's going to look like the absolute value of x minus 1 to the k, and then divided by k plus 2 times 4 to the k. So again, I'm only putting the absolute values on the pieces that it might affect, okay? Alright, well this is just a big fat mess, so I see a fraction over a fraction, so let's go ahead and multiply by the reciprocal of all of this stuff down here.  Alright, well this looks like a big mess too, which it kind of is. So let's go ahead and break it apart and kind of group like things together. So that's, we still got our limit, k goes to infinity, I don't want to lose that. So I've got my absolute value of x minus 1 to the k plus 1, and then oops, I lost my absolute value. Then in my denominator, I've got the absolute value of x minus 1 to the k, right? Those two pieces are like, and we're multiplying all this junk together, so it doesn't matter how we break it apart. Okay, I've got times, let's go for that k plus 2, and then I see a k plus 3 in my denominator, that's kind of similar. And then times, I've got my 4 to the k in the numerator, and then my 4 to the k plus 1 in the denominator. Alright, now we're getting somewhere. This is looking much nicer than I did before. Okay, so we've got the limit, k goes to infinity, and now we've got to use some properties of exponents here. So hopefully you remember that. And in this first piece, this x minus 1 part, we're going to be subtracting those exponents because we're dividing, so that's just going to leave us with an x minus 1 in absolute value. The second piece here, I can't really do much with that, so let's just rewrite that. And then the third piece here, we can definitely use our properties of exponents again, which is going to leave us with 1 fourth. Woohoo, alright, that's nice. Okay, so I'm going to box this area off over here, sorry. So we've got the 1 fourth and the x minus 1 no longer have anything to do with k's, so let's go ahead and take those out front.  Okay, so now using L'Hopital's or dividing each piece by k, however you want to look at it, hopefully you see that this limit just goes to 1, right? That's not so bad. So we've got the 1 fourth x minus 1 left, and remember I said this converges when all this stuff is less than 1. So that means that the absolute value of x minus 1 then must be less than 4. Okay, so make ourselves a number line. Stick 1 in here. We're going to be going 4 to the right, which leaves us at negative 3. Not a big surprise, that's what we saw in our graph. And if we go 4 to the right, that's going to take us to 5. And again, not too big of a surprise. Remember I said it looked like 4 between 4 and 5, something funky was going on? Okay, so now this is where we actually have to test these two values, and we need to see does our series converge or diverge at those specific values? Okay, all right, let me go down here. So we've got x equals negative 3. I'm going to plug that into our series, so that gives me the series k goes from 1 to infinity of, let's see, so remember it was x minus 1 to the k, so negative 3 minus 1 is going to give us a negative 4 to the k, all over k plus 2, 4 to the k. All right, again, let's do a little bit of algebra here. So hopefully you agree that I can break apart this negative 4 to the k and I can write it as negative 1 to the k times 4 to the k because I want to cancel this stuff in my denominator. Okay, so 4 to k's cancel. I end up with negative 1 to the k over k plus 2. This is an alternating series, obviously that's going to go to 0, so this thing converges. All right, so negative 3 were good, and again, looking at our graph, that made sense, right? At negative 3, it looked like they were all kind of clustered in there together. Okay, x equals 5. Oopsie, I don't know what happened there, let me try that again. So I've got my sum, k goes from 1 to infinity. I'm going to plug 5 in, so 5 minus 1 gives me 4 to the k, all over k plus 2, 4 to the k. Again, my 4 to the k's are going to cancel, so that's going to leave me with, oh jeez, my sigmas are getting worse and worse. k goes from 1 to infinity of 1 over k plus 2. Does this converge? Does this diverge? I don't know. It kind of looks like the series of 1 to the k, so let's use that limit comparison test. This is a good time to refresh some of these tests. There are lots of other tests you can use on this one, but this is the one I'm choosing to use. So I want to compare it with the series, k goes from 1 to infinity of 1 over k. That's a harmonic series, and that one's going to diverge, just for the record. So the limit comparison test says everything has to be positive. Check. Now we need to do the limit of, we're going to divide these guys out. So we want the limit, k goes to infinity of 1 over k plus 2 over 1 over k. Again, we got fractions over fractions, so we want to multiply by our reciprocal. I'm sure you guys can see this is a lot of algebra with this stuff. Okay, take the limit as k goes to infinity. Again, we've got k's in the numerator and the denominator, so this limit's going to go to 1. 1 is a positive finite number. Okay, so that's a good thing. So that means whatever the series we were comparing it to did, our original series does too. So this series diverges, so that means this series must diverge as well. Okay, which again makes sense because it looks like at 5 stuff shooting up to infinity. Okay, so our interval of convergence then is going to be negative 3, and I want to include that because that one did converge to 5, but I'm not going to include that, so I'm going to put a parenthesis there because that one diverged. Alright, so there's lots of pieces here with these, and you've got to put all those pieces together of stuff that we've done before. Alright, thank you for watching.


It's fan appreciation day here and we supported the fans and employees have supported the players for 71 home games. So now it's their turn to say thank you to the fans for supporting them for coming in, cheering them on. We got us regardless of the winning losses. Helping one another out, this is a lot of fun.


Hello, Psych2goers! Welcome back to our channel. Sleep. We all need it. A lot of us neglect it. And a lot of us have some interesting sleep habits. Some of us toss and turn in the night. Some of us are pure night owls. Some of us can't stop mysteriously waking up at 3 a.m. Ooh, something about that time gives me nightmares. Well, it would, I mean, if I was asleep. But did you know your sleep habits can tell a lot about you? Here are what your sleep habits say about you.   No? Okay, well, according to some studies, it can tell you a lot about your intelligence type. If you're starting to wonder now, morning people are more likely to get good grades in school than night owls, according to research studies on students in high school and college.  In the same study on high school and college students, night owls scored higher in general mental ability. This ability is linked to higher success in your career. So while you may not always get the best grades as compared to those morning people, you're still just as smart as them, perhaps just shown in different ways. Number three, you stay up late and deprive yourself of sleep. Staying up late again, binging another season of your favorite Netflix show up into the wee hours of the night with another psych-to-go video in a bowl of popcorn. Are you microwaving said popcorn as we speak? Many who often go to sleep late at night are often seen as pessimistic. Why? With the unpredictable and disrupted sleep patterns, negative thoughts are bound to be, well, abound. Maybe you find yourself up so late at night imagining possible scenarios to your weekly conflicts or ruminating on the latest worries that plague your mind. Whatever it is that's keeping you up, worries or television, it's best to hit the hay and get your Z's by going to bed as early as possible. Lights out. Number four, you always wake up between 3 and 5 a.m. no matter what time you went to bed. Do you find yourself naturally waking up early in the morning?  Well, this may have to do with your sleep drive. Part of our circadian rhythm, our sleep drive, grows as we go about our day. At night, it naturally goes down. If you find yourself waking up between 3 a.m. and 5 a.m. and don't know why, it may be that your sleep drive has been used up. If this becomes a common occurrence, it could mean you've developed a circadian disorder known as advanced sleep phase syndrome. Number five, when you don't set an alarm, you sleep for 14 hours at a time. Finding yourself sleeping more than the regular 8 hours at a time, maybe you find you naturally sleep through your morning or half the day. Well, one reason could be hypothyroidism. This most often affects women over 60. This deficiency of thyroid hormones can disrupt one's heart rate, metabolism, and body temperature. The symptoms can include fatigue, weight gain, cold sensitivity, constipation, and dry skin. Another possible reason you're oversleeping? Your body could be responding to a latent infection. But if oversleeping becomes the norm, something else could be going on. Number six, you're tired every morning. Do you have this problem? No matter how many Zs you seem to get, you just can't help but wake up tired every morning. One reason could be depression. Depression often coexists with sleep problems. So if you're tired every morning, a mood disorder may be the reason. Another reason? You may just have sleep apnea. This sleep disorder can be serious and it affects your breathing during sleep. One's breathing repeatedly starts and stops when asleep with this disorder. Your sleep cycle is likely to be interrupted with all these interruptions in breathing. If you feel you may have sleep apnea, you're not alone. The disorder affects 30 million Americans. A good idea is to sleep on your back instead of your side. If the problem still persists, it's best to talk to your doctor. Number seven, you sleep longer than you're used to. Okay, so maybe you don't sleep for 14 hours stretches, but you have noticed you keep sleeping a little longer and longer every night. One study on adults 40 and older found that individuals who slept more than the regular eight hours at night were more likely to suffer from a stroke. Researchers from the University of Cambridge found the risk grew notably higher for those who used to sleep for shorter hours. They're unsure if this is a symptom of a problem or if the long hours of sleep are the cause of other health problems. Number eight, you often sleep on your side. Although further research is needed, sleeping on your side may help in circulating waste out of your brain and reduce the chance of getting Alzheimer's. Researchers at New York's Stony Brook University conducted experiments on rodents and found that sleeping on one side makes it easier for the body to circulate waste out of our precious brains. Many believe Alzheimer's is linked to brain waste buildup, and since sleeping on your side may help your brain circulate waste, doing so may lower your chances of developing neurological diseases such as Alzheimer's and Parkinson's. Number nine, no matter how hard you try, you just can't fall asleep. Do you lie down in bed early in hopes of a good night's rest only to find that you just can't fall asleep no matter what you do? Well, as suggested before, this may be due to worry, stress, or work. Phillips Respironics conducted a worldwide survey and found that over half of their respondents reported that the main cause of their sleep problem was work or worrying over their finances. Maybe you can't stop thinking about what's worrying you lately or maybe you're working through the night over hours. Whatever it is keeping you up at night, it's time to put it to bed so you can go to bed too. So which sleep habits can you relate to? What did they say about you? Share with us in the comment section down below. If you enjoyed this video, make sure to like and subscribe to Psych2Go and hit the notification bell icon for more videos like this. And as always, thanks for watching.


I think over the last five years there's been very material progress in improving diversity. There's no doubt that it's accelerated in the last 24 months and there's equally no doubt that the Davis report, particularly from a gender perspective, has helped that acceleration and the adoption of the Financial Reporting Council to ensure that companies actually report on the diversity as added to the development and the momentum of progress. But overall good progress made and the numbers are starting to tell us that, particularly in the gender area. There has obviously been a lot of progress in driving diversity in UK boardrooms over the last five years. We can see some encouraging data in Lord Davis report, but a lot remains to be done. But if we look at the report and we'll see what have happened, 25% of all the new recruitments to boards the last year has actually been women. And if we look at the all-male boards, they have been reduced from, I believe, 21 to 11. So there is progress. And I think this also shows that the UK model of work with voluntary initiatives by the companies and avoid quotas. The speed of change to increase diversity in the UK boardrooms. Well, the great news we learned about today is that it's speeded up dramatically in the last two years, probably because of the enormous amount of focus and publicity that's been given the whole question. There are some certain facts and figures that have been given us, one of which surprised me. There are only 8% of the footsie 100 that don't have the representation of females on their board now, which is pretty dramatic. I think in the past five years it's been a bit mixed, to be honest, because I think we've made the most progress in the last 12, probably six months in many ways, certainly the last 18 months. I think before that we'd stalled. I think diversity generally is an agenda item during a recession. It's easy perhaps for boards to say we've got other priorities. And I think the 30% club has been very vocal in saying, actually, no, let's not put this on the back burner. Let's keep this uppermost on the agendas of boards everywhere, particularly in the UK.


We often will find that the Chief Strategy Officer is responsible for defining the direction of the organization and then there's a handover to the delivery side or the execution of transformation teams whose role it is then to make that idea a reality. You should make sure you set your strategy expectations realistically because they're always grounded in your current capability the way you run your company now. And big strategic changes require big change management programs. It's cascading down the organization a way of talking about your strategy, setting out what strategy needs to look like, and then having the dialogue that helps these business levels help one another own and refine their strategies and make sure they fit together and reinforce each other. That is a tough job, but that is execution. That's taking accountability for execution. Many leaders fail to understand or underestimate how much efforts, how long it takes to really build the systems, the processes, the procedures, the needed talent base and change the culture in order to implement the strategies. I think it really starts with engagement. People need to feel engaged about their roles when it comes to delivering the strategy. What we've seen that doesn't work is a very top-down approach, a very authoritative approach. People are given instructions, but they're not really engaged about what is their role in terms of delivering key outputs for the organization. I think a distinction that Jeff Bezos makes in the way that decisions are made is very valuable with respect to how you speed up decision making. He says, you know, there are two kinds of decisions. There's what he calls type one decisions. Type one decisions are risky, irreversible, they commit to the organization, they can bring the organization's brand to a bad place, and they require a fairly substantial public commitment. Type two decisions in contrast are easy to make, they're reversible, they're inexpensive, they're low-risk, they are rich with learning, they can be done quickly. And for those kinds of decisions, you know, a small team should be empowered to go and try stuff. Don't bring people to the work. Bring the work to the teams. Make sure that the work is defined in a way that the teams can be successful, that links clearly to our strategic purpose, and ensure that they have the tools and techniques and support they need in order to deliver successfully. The bigger problem, though, is that we need to balance this natural bias for efficiency and execution with an equal focus on creativity and flexibility and experimentation to create tomorrow's business. And that involves some sacrifice of efficiency for the sake of future option value. And that sounds like a contradiction and it is, which is why ambidexterity is so hard. If you're going to add in new initiatives, then be very thoughtful about what it means for the initiatives that you already have in play and the demands that you're placing on your people. Once again, you know, we all too often, in theory, assume that there is an infinite pool of highly capable people available to deliver the strategic initiatives. It's a finite resource and therefore it has to be managed in a very definitive way. Keep the cost of failure cheap. Then you can fail more often. For me, failure is nothing more than converting hypothesis into knowledge. That's all failures.  Therefore, it is possible to de-risk. I'm not asking any company to throw billions without thinking because the future is inherently something you can't predict. But you must be prepared to proceed in a future that you can't predict. So imagine if we change the criteria and we said, we celebrate people that find out wrong things quickly. Like the quicker you learn something is not going to work and you show it to us, you're the best employee we have. And when that happens, it encourages people to then quickly find stuff out whether it works, whether it doesn't work and then quickly share those learnings because they know that that will be celebrated rather than just showing success. It's time for you to join these leaders, register now and turn your ideas into real results.


Hello, and welcome to this presentation of the STM32L0 series of Ultralow Power Microcontrollers with the 32-bit ARM Cortex M0 Plus Core. The STM32L0 series is part of ST's 10-year Product Longevity Commitment Program for STM32 and STM8 microcontrollers. To break ultralow power challenges, battery-powered applications need four key ingredients. Low consumption. Ultralow power applications need to use devices with the lowest power consumption. The STM32L0 is based on ST's own ultralow power process technology and it provides record-breaking low power performance. Energy efficiency. The level of efficiency is one of the key elements that contribute to the success of ultralow power applications. From the heart of its architecture right to the end of its peripherals, the STM32L0 comes with innovative features allowing to conserve any small amount of energy and to reach outstanding energy efficiency. Maximum integration. On top of low power, another key element is the need of high integration. STM32L0 is available in tiny packages with large memory size, various set of peripherals, and full security features. Cost savings. Last but not least, a successful solution has to be cost-effective. The STM32L0 has a competitive portfolio and comes with a free software development ecosystem. The STM32L0 series offers seven different power modes to fit any power-sensitive application profile. For applications that spend most of their time sleeping, the STM32L0 offers dedicated stop and standby modes. Standby mode is designed to save the maximum amount of energy, shutting off all that is not needed. The wake-up time from standby to active mode is only 50 microseconds via a reset or by one of the two wake-up pins. Stop mode is very flexible to perfectly fit any application. With the RTC enabled, providing time counting, sub-second counting, auto wake-up, calibration and calendar features, stop mode consumes only 5.4 microamps under the highest temperature condition at 125 degrees Celsius. STM32L0 devices can wake up from stop to active mode using up to 16 wake-up lines in only 3.5 microseconds from RAM or 5 microseconds from flash memory. To optimize power consumption while sleeping, the STM32L0 provides a low-power sleep mode that consumes only 3.6 microamps with the flash memory off and 14.5 microamps keeping the flash memory on at ambient temperature. For applications that require very low active current versus efficiency, here again, the STM32L answers the call with its low-power run mode, which consumes as little as 18.5 microamps. At full speed at 32 MHz, the STM32L0 sleep mode consumes only 37 microamps per MHz and active mode is optimized to consume just 139 microamps per MHz at full speed. On top of that, the STM32L0 is, in its category, the undisputed market champion at 125 degrees Celsius with the lowest current consumption and the only one sustaining such high temperature in the industry. This graph shows the position of the STM32L0 series platform versus comparable competitor devices. It shows the evolution of the lowest power mode with RAM retention versus the temperature and the device's temperature limitations. The result is clear. The STM32L0 is the only device functional up to plus 125 degrees Celsius with the lowest current consumption. Healthcare applications such as fitness trackers, blood pressure gauges, or glucose meters need to combine rich analog features together with an optimized battery lifetime and competitive solution. The STM32L0 is a perfect match to answer healthcare needs. The STM32L0 comes with rich analog features. Its 12-bit ADC is optimized to consume as little as possible. It can operate under a low voltage going down to 1.65 volts and in terms of current consumption, at 100 kilo samples per second it draws only 48 microamps from the power source. The ADC is not only able to reduce the energy cost to acquire analog signals it can also achieve high resolution up to 16 bits by hardware oversampling. To complete the analog feature set, the STM32L0 integrates a 12-bit DAC and two built-in comparators. Moreover, the STM32L0 is able to minimize the inrush current that protects the battery and maximizes its lifetime. The STM32L0 is the right device to meet the challenges of applications such as IoT, remote controllers, and accessories. Such applications are very sensitive to bomb cost, PCB size, and battery lifetime maximization. To reduce power consumption, the STM32L0 series hosts a low-power UART with wake-up capability as well as a low-power 16-bit time counter. Thanks to the ultra-low power timer, users can keep counting time in stop mode using a power consumption of only 440 nanoamps with full RAM retention. To help customers find the best design in regards to bomb cost, the STM32L0 series offers a built-in LCD driver and a USB 2.0 crystal-less interface that supports battery-charging detection and link power management features. No compromise on CPU speed, the STM32L0 offers a wide-voltage range with full speed capability. Therefore, no need to increase VDD to run at high speed. Now, ultra-low power is synonymous with high-processing speed. The STM32L0 shows high integration with large memory size with up to 192 kilobytes of dual-bank flash memory and 20 kilobytes of SRAM. Moreover, it provides a full security feature set, including hardware encryption capability, flash protection, true random number generator and a 96-bit unique ID. These hardware features combined with the STM32 ecosystem makes the STM32L0 ready to be class B compliant. These MCUs are available in small sizes down to 2x2 mm for WLCSP packages. The STM32L0 series is available in three different lines, which share a common architecture with five clock sources, three internal and two external, USART, SPI and I2C communication peripherals, DMA, 12 or 16-bit ADCs, comparators and temperature sensors. The STM32L0X1 is the access line. It is designed to answer the needs for entry-level ultra-low power applications. The offer goes from 14 up to 100 pins and from 8 up to 192 kilobytes of dual-bank flash memory with 20 kilobytes of SRAM and 6 kilobytes of EEPROM. The STM32L0X2 USB line extends the access line and features a 12-bit DAC, 16 capacitive touch keys, a true random number generator and USB 2.0 FS crystal-less options. The STM32L0X3 USB and LCD line extends the USB line by offering a built-in LCD driver. The three STM32L0 lines offer optional 128-bit AES hardware encryption if needed. The STM32L0 series includes more than 100 part numbers, providing a large choice of memory sizes, pin counts and packages. Thanks to this large product portfolio, customers can find the right product tailored to their needs. With the unit price as low as a single 8-bit device, switching to a 32-bit one has never been so easy. A 32-bit performance at an 8-bit price without compromise on power consumption and robustness. That's the deal. The STM32L0 is a great investment. It benefits from pen-to-pen compatibility across the STM32 family. The STM32 portfolio offers an extraordinary variety of compatible microcontrollers based on ARM Cortex-M cores. M0, M0+, M3, M4 and M7, giving developers flexibility to find the perfect STM32MCU for their applications. The STM32L0 series offers three different kinds of boards to evaluate, discover and play with the series. While the evaluation board will let you study in-depth the STM32L0 platform, the Discovery Kit, with its e-paper display, is more application-oriented. The STM32Nucleo board with its Arduino connector is dedicated more for fast prototyping. The STM32Nucleo board is embed-compliant like any other STM32Nucleo board. Thanks to STM32CubeMX, users can optimize the layout, configure the MCU and evaluate its power consumption in their application. The STM32CubeMX solution also facilitates the migration from an STM32L0 to any other STM32MCU series. To complete this offer, the STM32L0 series is supported by free IDE solutions such as Keele, AC6 system workbench and KuCox. The highly affordable STM32Nucleo boards allow anyone to try out new ideas and to quickly create prototypes with any STM32MCU. Four Nucleo boards based on STM32L0 microcontrollers are available. STM32Nucleo boards can easily be extended with a large number of specialized application hardware add-ons, Arduino, Uno, Rev3 and ST Morpho connectors on Nucleo64, Arduino Nano connectors on Nucleo32. It has never been so easy to select, download, watch tutorials or get the latest news about STM32 products and solutions on a computer, tablet or smartphone. The STMCU Finder app lets you quickly and easily select the best devices for your application on a tablet or smartphone. Visit www.st.com slash STMCU Finder for more information. The ST microcontroller forums allow the STM32 developer community to discuss and share experiences. If you would like to know more about STM32 products and solutions in real time, join the growing STM32 community on Facebook. For more information, please visit www.st.com slash STM32L0. Thank you for your attention.