Welcome back everyone. Now we're ready to talk about the ratio test of all of the convergence tests that we've computed in this lecture series I would say that the ratio test is probably my favorite of all of them So it's sort of like the best for less right here and the ratio test gets its name for the following reason I want you to consider the sequence of ratios. So you have some sequence a sub n like you see right here I want you to take the ratio of consecutive terms So take the term a n and divide it from its successor a n plus 1 or if you want to think of it take Take a term in the sequence and divided by its predecessor So take the ratio sequence of consecutive terms also take absolute value because we don't really care about positive or negative right here and So take the limit of the ratio So take the limit as n goes to infinity of a n plus 1 over a n all inside absolute value Let's say that this sequence is convergence. So the sequence of ratios is converges to some limit value call it L So that's how the ratio test gets its name. We have to compute the limit of this ratio Now if that limit is less than 1 then we can conclude by the ratio test that the series n equals 1 to infinity the sum of a sub n is Absolutely convergent now remember what absolute convergence here means it means that if you take the sum of Absolute values, that's a convergent series But as we've seen with the test of absolute convergence that if a series is absolute convergent then it's can its convergence as well And so the ratio test doesn't just give us convergence It actually gives us absolute convergence if this limit is less than 1 Now be aware that we get we can get away with absolute absolute convergence here because again We're kind of taking absolute values It's built into the mechanism of the ratio test here and this limit right here is basically measuring how close together Are the terms in the sequence how rapidly is the sequence of ratios shrinking because if this thing is shrinking Rapidly enough then its limit will be something less than one I should say that if the terms in the sequence are close enough together and That's going to give us that this sum right here will be convergent because the terms a n are shrinking at a wrap-enough rate That's that's what we're going to get from this limit right here Now on the other hand if the limit is greater than one if their limit of the ratios is greater than one Then this tells us that the series is in fact Divergent right it'll be a divergent series and the other possibility is if the limit of ratios is equal to 1 Then the ratio test is actually inconclusive it turns out in that situation the series could be absolutely convergent It could be convergent, but not absolutely that is it could be conditionally convergent or it actually could be divergent I'll show you some examples of that in just a second But let me show you some affirmative situations first so if we're going to test If we're going to we're going to try the testing the convergence of a series We'll use the ratio test here take the sum or n equals one to infinity of the alternating series negative one to the end You have n cubed over three to the end now you'll notice here some things going on here There's this geometric part three to the end, but this is not a geometric series There's this like in cube floating around, but this is not a p-series There's also this negative one to the end. It is actually an alternate series So one could try to determine whether this thing is convergent by the alternate series test Which you could be successful in doing that but one advantage of using the ratio test here is that the alternate series test can only show Convergence, but the ratio test can show Absolute convergence and then frankly it might actually be able to do it in an easier fashion So how does one actually work this thing out? We want to consider the the ratio sequence take a sub n divided by a n and so oftentimes what happens is that the terms in your sequence are actually fractions So when I actually work at this sometimes I actually want to take the a n plus one term and then kind of separate Separate it as a one over a n term because these terms are often both fractions themselves So the a n plus one is going to look like the following We have negative one to the n plus one power. We're gonna get in plus one Let me rewrite that we're gonna get in plus one cubed all over Three to the n plus one like so that's the first term and then the next term is gonna look like taking the reciprocal We get three in on top and then we're gonna get a negative one to the n times n cubed like so And so we want to simplify this thing That's the next thing you have to do is you have to simplify this expression now some things to mention here Is that since you're taking absolute values any factors of negative one can just be entirely ignored It does not matter whether or pods are negative those negative signs will always disappear with the absolute value The next thing I want to mention is that you want to rewrite this fraction putting things of similar type together Like notice how there's this exponential term three to the n and there's this three to the n plus one I want you to rewrite it so that those two friends are now together So we have three to the n over three to the n plus one and then the other terms We have this n plus one cubed and this n cubed those are kind of the same type put them together as your next fraction You're gonna get in plus one cubed over n cubed Like so and again now we're gonna try to simplify these things Because when you look at three in plus one on the bottom here This thing can break apart as three to the n times three and so this three the n cancels with that three to the End and you're left with the absolute value as one third, which is actually just a one third right here Now with the second bit in plus one cubed over n cubed both the numerator and denominator are being cubed And so if we bring out the exponential we're going to end up with n plus one over n cubed Are we writing it one more time we get one third we're going to get one plus one over n quantity cubed here And so in this situation as we take the limit as n goes to infinity because again That's what we're trying to do here. We want to take the limit All right, so this is a limit calculation the limit as n goes to infinity If you carry that through all of these calculations we get to the very end here We take the limit as n goes to infinity Well as n goes to infinity The one third will stay one third the plus one will stay there this one over n part will vanish to zero And so we end up with this this this sequence of ratios will converge towards one third times one cubed And which of course that would then just become the number one third like so This is the limit l that we saw at the ratio test. This is the limit of the ratios All right, and so because it's the limit of the ratios We want to compare with the ratio test the ratio test that we bring it back over here If our limit is less than one this says the series is absolutely convergent And that's exactly what we see over here This limit is less than one and so then we conclude by the ratio test That our series is absolutely Convergent and since it's absolutely convergent that actually implies it's convergent So if you're asked is the series convergent you would say yes It's convergent if you want if you could also further specify It's absolutely convergent and we use the ratio test in so doing As another example, let's take the let's take the sum of n equals 1 to infinity of n to the n over n factorial Now, this is not an alternating series So we couldn't use the alternating series test This in factorial makes it very difficult to integrate because what's the anti derivative of x factorial e? You know, that's that's kind of a weird thing that that's not a continuous function that we can find an anti derivative of But the ratio test actually works out really great in this situation Because our ratio test again, we're looking at the ratio of a n plus 1 over a n This is going to look like n plus 1 to the n plus 1 power over n plus 1 factorial That's the a n plus 1 term and then if you look at the 1 over a n term That's going to be in factorial over into the n And like we did last time put together similar types like we're going to put the factorials together And then put the exponential expressions together as well if we do the factorials We end up with n factorial over n plus 1 factorial And that's going to be paired up with the n plus 1 to the n plus 1 over n to the n like so All right Now much like we've done the previous example these ratios have to be simplified in some manner Now n plus 1 factorial has a very nice factorization as a factorial. This will factors n plus 1 times n factorial for which the n factorials then cancel right here And then another sort of curious thing here is that you have an n plus 1 The n plus 1 that's in the denominator can actually cancel with one of the n plus ones that are over here And so this ratio simplifies to be n plus 1 to the n above n to the n And much like our last example since we now have They're both to the power of n we can bring that out of our fraction and get n plus 1 over n To the n right here Uh as simplifying this we're going to get 1 plus 1 over n to the n now this one's a little bit more tricky um because we can't if we just set in to infinity right you're going to end up with uh this thing will look like This thing would look like 1 to the infinity which is actually in determined form So this kind of leads to a lopital argument For which if you want you can see the the video link that's in front of the screen right now You can actually see why this calculation turns out the way it is But this limit will turn out to be the number e Which is approximately 2.7, etc, etc This is our limit value and it is actually greater than 1 So what this tells us by the Ratio test the ratio test would give that this series is going to be divergent This is a divergent series by the ratio test And so when we go back to the ratio test we saw here if our limit's greater than 1 the series is divergent We're using that as we're using that evidence right here. Now I should mention that in this last example um, we found out that the limit The the limit of the an plus ones over an's to get absolute value there if this limit Goes off towards something some l value, which is bigger than 1 It turns out you actually didn't need to use the ratio test whatsoever you actually could have gotten away with the test for divergence or the so-called divergence test The test for divergence The reason that this limit goes to something bigger than 1 really comes from the fact that the sequence a sub n Does not go towards zero and if you look at the proof of the ratio test, that's why that's happening So we actually could have used the divergence test in that situation Because again going back to our sequence and play here this n to the n over in In the factorial here This isn't an obvious fact to us perhaps But the thing is the top is growing a faster rate than in factorial that this thing actually converges towards infinity Right. It's not the fastest growing function necessarily, but ultimately into the end The power exponential function beats the factorial so we could have used the divergence test there as well But you know, that's the thing about the ratio test is that if it turns out that you get in this situation You still get an answer, right? You don't just have to all your effort isn't wasted It's like I could have used the divergence test But since I didn't use it and I went to the ratio test I still get an answer as opposed to like the alternating series test Which it can tell you when the series is convergent But if if the alternating series test fails that doesn't imply convergence You'll need you actually have to do the divergence test in that situation I want to talk about at the last part of our video here. I want to talk about this last case a little bit. What happens? What about when the ratio test is inconclusive, right? Well, I mean if it's inconclusive that means you need to do something else But why do you need to do something else here? If it's inconclusive? We can draw no conclusions because of the following type of situations If you take if you take a first example here Take the sum of 1 over n squared, right in this situation This series is convergent. It's a p-series. In fact, it's absolutely convergent Because it's just a positive series there. That's convergent But if you look at the the ratio sequence a n plus 1 over a n This thing is going to look like n plus 1 squared over n squared and that all converge towards the number one. It's an absolutely convergent series, but um But the limit goes to one so if that limit ratio limit equals one it could be that it's uh, Absolutely convergent. It could be another option Take something like the following take the harmonic series 1 over n In this situation, we know that the harmonic series is divergent But if you look at the ratio sequence a n plus 1 over a n This thing that's going to look like a n plus 1 over n Which is that likewise converges to one so the limit when it's one it could be absolutely convergent, which is convergent But it could also be divergent and it turns out there is a third possibility, right that If you take the sum of like say the alternating harmonic series a n A negative 1 to the n plus 1 times 1 over n. This is actually an example of a conditionally convergent series But in that situation if you take a n plus 1 over a n You still end up with Just the same thing you did here You're going to end up with a n plus 1 over n because the absolute value is just negate the alternating fact This is going to go to one So you can see that when When this limit of ratios equals one we truly are inconclusive because you can absolutely convergent You can get divergent and you can get conditionally convergent Now one thing I should mention is that the conditionally convergent situation must occur in this case where the limit goes off to one But we could actually determine divergence or absolute convergence if the limit is greater than or less than one depends on the situation But be aware that if the limit of the ratios goes to one you have to use a different Test than the ratio test. So although I think the ratio test is super awesome It does of course have some limitations Particularly this this inconclusive case when the limit goes to one


Okay, we're back live here in Silicon Valley, California for Brocade's special tech analyst day. This is SiliconANGLE.tv's theCUBE. I'm John Furrier, the founder of SiliconANGLE, and I'm joined with Stu Miniman from Wikibon.org, and we have the CTO Dave Stevens VP of Corporate Development as well. I am. So you're in charge of all the M&A and the money and acquisitions and big investment decisions. Exactly. So you got it within some constraints, no doubt. So you guys made a good investment a couple of years ago in this whole idea of software and the network fabric. Sure. Take us through one, what's happening today, and take us back to that investment decision and how it all came today. Well, I think the investment decision really happened back, I think we started the development back in 2008. We had done very well in the storage networking business, and we saw a lot of the characteristics of the storage networking business and what customers liked about that, the resiliency, the efficiency of those networks, and we were looking to take that discipline and take that engineering, the technologies into other areas of the network, and we saw the emergence of virtualization technologies inside of the data center. What was happening there and what was very clear was that there was too much complexity in the Ethernet side of the data center, and so we made the decision to take that same resiliency in that flat network architecture that had had so much success on the fiber channel side of the network and apply that over to the Ethernet world. So we started that development back in 2008. And now you got the fabrics and the Ethernet fabric out there, and now we're just at VMworld, where we saw the huge buzz around this era, and the home market is going crazy for network virtualization. And so here we are, right, going the next massive growth. So how do you guys feel about that, and what's your mindset right now? Well, I think we are fully supportive of all the emerging software-defined networking technologies and network virtualization. I think the requirements out of customer environments are very clear. Customers want to take complexity out of that data center infrastructure, and ultimately what they want to do is they want to solve business problems. So they want to look at their data center infrastructure as a pool of different assets. They want to look at security elements. They want to look at load balancers. They want to look at virtual machines and applications. They want to look at pooled storage. They want to identify those resources, and then they want to effectively assemble those into a software-defined data center. And the way that you have to do that is to take the network infrastructure that's in place in the data center, and you have to add an ability externally to connect up those resources on command. And that means you have to add a programmatic interface and a software-defined interface to the existing network infrastructure to make that happen. So that's where the customers are taking us. So Dave, the word of the day is simplification, automation, and innovation I think we're talking here.  Was that kind of just a basic philosophy, or is it something that you kind of consider and are looking along the way as to what happened? Well, I think as you know about us, we use a combination of merchant silicon in certain products within the portfolio, and then we invest and use ASICs in other parts of the portfolio. And I think it's very well understood that if you can correctly identify the specific application, if you know what the requirements are, you know what you're trying to get done. You can build a tremendous amount of technology on a silicon. You can get to better power levels. You can get to higher levels of reliability. You can get to better performance levels, but it's a major investment to go do that. I think the merchant silicon vendors in particular have to serve a much broader audience of applications and a much broader audience of consumers. In the data center in particular, we know exactly what we want to get done. We have the teams in place to go do that, and so we can apply ASIC technologies to those issues, and we can get to products that we can't build any other way. When it makes sense, we invest in the ASICs and other places we continue to use merchant silicon. So it is a significant investment. How do you make those decisions as to which markets you're really going to target and put the functionality, and how do you stay? The development cycles are long, so walk us through that process. Yeah, well, I mean, we have a team today that's capable of doing multiple large capacity ASICs. We try and stay on the absolute leading edge of a process technology so that we get the best efficiency that we can get. Again, we target environments where we know specifically what the requirements are and where the products can benefit from the investment. If it's a more general-purpose product, we probably won't use ASICs. When it's a product in the data center, which is really our home turf, then we'll use ASICs in there, and we'll put those into specific applications where we know exactly what's happened. In the fiber channel space, for example, we're shipping, I believe, our seventh generation switching ASIC products today. We rolled out four or five more ASICs today and some of the product announcements that we did around the VCX, the VDX products in the data center, and also accelerating and improving the 10-gig density on some of our MPLS routers. I know we're not talking much about fiber channel at the show here. It's a real focus on ethernet, but let's talk about fiber channel for a second because I've seen a good amount of brocade solutions ending up in converged solutions. We were at the EMCV specs launch.  Hatachi is using you. Wikibon, we did kind of deep research on convergence, and we're seeing that it's just at the beginning of this trend, whether it be single skew environments or the reference architectures, what's your take on where convergence is today? Well, I think where convergence is being applied in the network today is where it makes sense to do it. So at the very edge of the network, where you typically have servers with a lot of adapters in them, a lot of fiber channel adapters, a lot of ethernet adapters, you have an equivalent number of switch ports as the first hop into the network. That's a location where it makes a lot of sense to take storage traffic, take ethernet traffic, converge those onto a pair of CNAs, run a couple of cables out of your server, come into the first hop in the network. I apologize. Not speaking specifically about convergence, same with FCOE, but talking about stack convergence, building environments where the networking is part of the whole solution, whether it be something like IBM where a box and they stick your blade in, or it could be specs where it's your box and a reference architecture. So the whole selling that kind of single, tested out, baked environment, I guess I brought up questions, where does networking sit in the stack and what's the evolution of that? Well I think you never get rid of the networking component because it is the essential element that connects everything together and provides connectivity between all the different elements. I think there is a desire among the customer base to buy pre-configured tested stacks and to buy those in pods, but at the same time, customers want the flexibility and they want the ability to use different vendors and use different components and get the best or break components within those stacks. And so it's a buying convenience and it's a testing convenience to buy them a package at a time, but they don't want to give up the flexibility to change out of that package and go to a different vendor as necessary. To the extent that there's better integration by pre-packaging things and pre-integrating things and it takes out some of the configuration nightmares that customers have, I think that the industry will continue to move that direction. Dave, I want to ask you about the Nassir acquisition because obviously that's causing a lot of noise in the industry and buzz, buzz and noise, it's good, it's good for everybody and it's real pushing the envelope, we love it, but that being said, obviously a big statement by VMware and the traditional network vendors, you guys, Juniper, Cisco, mainly Juniper and Cisco are all going wow, as much as they don't say oh we're not affected by this and VMware put out some sort of press release that they're partnering with Cisco. It wasn't really nothing in an ounce other than they're partnering with Cisco. It catches people off guard, so we've seen kind of a repositioning around existing networking. Martine from Nassir clearly said on the queue, hey I want to disrupt networking, that was his passion, it's being disrupted, so how do you see this all evolving relative to the incumbent networking guys? We'll throw HP into that mix too, they got to essentially think, hey the data center's changing, we're seeing clear examples of that, solid state is changing caching, architectures anywhere across the line, the role of data, and networking has been that last bottleneck that's now being addressed by virtualization, so as Mike said, it's a decade's worth of build out and we're at the beginning. I think there's a very clear desire among the customer base to simplify the networking architecture and simplify the connectivity of the different pieces within the data center. Ultimately what customers want to do is they want to select resources in the data center and then they want to have the network provide connectivity between these resources in the data center and give them the ability to effectively build a virtual data center driven by software. One way to begin doing that is to simplify the architecture that you're building on top of. So this is really our focus around fabric technologies to take what customers clearly don't like, which are these deep, hierarchical complex networks with lots of proprietary protocols and move those to a flatter, faster architecture that's built for virtualization and built for multi-tenancy effectively. What SDN effectively adds on top of that is a software abstraction layer that gives us the ability to add programmatic controls on top of that network infrastructure. So in our view, the two work hand in hand. We're a big proponent of SDN, we're a big proponent of NYSERA and of Martins. We've had a long relationship with those guys as we have had with VMware and with Microsoft and our job is to make the network infrastructure as efficient as possible and make sure that it fits into those SDN environments as they evolve. I think as you saw it, part of Analyst Day today, we've done some very comprehensive announcements with VMware, for example, around VXLAN and around NVGRE, as well as the NYSERA controller. So we're all in. So Dave, when you look at what NYSERA is doing, one of the big pieces of it is distributed systems. So I heard two things that I wonder if you can comment on from your announcement today.  So, you know, I- Except for the latency and the speed of light problem, right? Yeah, latency and the speed of light, you know, I think- There's still a couple of challenges left there. So can you just talk to us a little bit about what you're seeing, distributed systems, whether that be kind of large data centers, service providers to the WAN, what are the biggest challenges you guys are facing and how are you addressing it? Yeah, well I think the biggest thing used to be, when we all sort of grew up in the networking industry, we were taught to keep as much traffic inside of the building as possible and minimize the amount that got transported across the wide area. And that's because you had to pay for the wide area bandwidth, where the bandwidth inside of the building was largely viewed to be free. That disparity in cost and performance levels is beginning to go away, right? I mean, you still have to pay for your WAN bandwidth, but today you can buy an awful lot of bandwidth for not very much money, and so it's making it possible to begin to distribute applications geographically between data centers. If you're going to do that from a networking perspective, you still have to overcome the effects of latency and the effects of distance in that kind of activity, but it is becoming possible to begin to distribute applications geographically and put them in the locations that it makes the most sense from a business perspective. So the other thing when we look at, I'm sorry, are you saying, okay the other thing when we talk about VXLAN is, if you look at kind of the layer four through seven applications, what lives in an appliance, what's the role of the F5s and riverbeds of the world, versus software and what's brocades placed in this whole ecosystem? Well, I think anytime you build out a data center, there are elements that you build out inside of the data center. You compute elements, you're storage elements, there's services elements like firewalls, load balancers, right? And those can be deployed in the data center either as a fixed appliance or they can be deployed increasingly as a virtual machine. And it really depends on the customer environment. There's certain customers that will deploy all those services as virtual machines. There's certain customers that build those as large appliances. And so I think both of those approaches are going to be around for a long time and both of them have to be accommodated within the data center architectures. Dave, I want to ask you a question around how tight on time and your handlers are wanting to move you to another location here this brocade tech day. But I want to ask you to talk to the audience about something kind of more abstract around the networking because with the iPhone 5 announcement today, obviously everyone's a buzz about the iPhone. It's just a great product and it really points to the pressure of the marketplace, the demand for this new era, new user experience and it's driving a ton of under the covers or if you look under the hood, it's just driving more and more traffic, obviously mobile. So on the consumer side it's pretty obvious there's a huge pressure on the service providers at every level, right? So that's one area, the other area I want, we could go to that in a second, but I want you to look down the enterprise because there you've got to push the envelope now and bring your own device to work has been talked about, but really there's new services, so there's like an iPhone effect going on at the enterprise. So talk about that and then also talk about the consumer. Where's the big pressure on the networking side that just wasn't around eight years ago? Yeah, well I think that the networking environment and the focus of CIO has moved from automating business functions. What Jeffrey Moore had typically called systems of record, right? HR applications, finance, accounting applications, things that got built inside a data center. We've now with devices like the iPhone 5 and the iPad and the Galaxy tablet and other types of devices out at the edge of the network we put so much processing capacity out in the hands of the average user that the focus of the CIO has now changed, right? If I'm sitting out with my iPad I'm effectively my own IT department. I pull information from colleagues, from competitors, from third party websites from my internal website and I amalgamate that information up on my iPad I've solved my own business problems and I make my own decisions out there and there's obviously that and BYOD causes all kinds of problems for the CIO around security but from a networking perspective it completely changes the dynamics of the network because now I have an originator and a consumer of information that is completely mobile, that is completely unpredictable and the sources that they're going to use to derive their information there's no way to actually predict where it comes and so the traffic patterns within the wider network infrastructure have become completely unpredictable and the only thing we can really do to accommodate that is try and build sort of the next generation of global communications infrastructure and build it in a way that's effectively limitless that facilitates connectivity from any user to any other user. So total disruption under the hood, complete re-architecture to make that happen and enable those new services. And we have to go rebuild it over the next decade. Final question I know I got a short answer because you got to go. What investment areas are you looking at right now? Obviously you guys made a good call, good investment for Procade. You're in a good spot, market spun right where you thought it would go. You skated through where the puck is. What's your investment focus right now in the Corp Dev and just overall corporate-wise? Yeah, well I would say we continue to focus where the company was born which is in the data center environments we have unique expertise that started there with the fiber channel space but it's all around six nines of reliability and power efficiency and things like that. So one focus is taking that philosophy to the other parts of the data center predominantly with either net fabric technology but also with core routers and traditional top-arack switches and application delivery controllers and other kinds of technology. I would say our other focus area is where there's tremendous spending and pressure today which is in the service provider market. The service providers as a whole are carrying more traffic. They're not getting really paid for a lot of the traffic. There's a cost pressure in those environments and they're re-architecting the carrier networks to be able to carry more applications and more traffic and do it at lower cost and that's causing them to re-architect and look at new technologies in those environments. And ultimately those carrier networks are what connect the two data centers together. So those are the two main focus areas that we're working on. Dave, Steve is CTO VP of Corporate Development Brocade Innovation Day here talking about what's happening under the hood, powering all the mobility, the iPhone 5, enterprise applications, be your own IT department. Great for this new era. Thanks so much for joining theCUBE. We'll be right back with our next guest on SiliconANGLE TV right after this short break. Thanks a lot.