 Welcome back everyone. This will be our last video for lecture 35, which turned out to be on the little bit of the long side here We've seen previously How with a sequence there is no notion of tangent line, but we can still capture some ideas about optimization and monotonicity for sequences sometimes we can't borrow the derivative if the function can be Extended in a continuous differentiable way But we also seen that sometimes that the sequence approach can actually be more efficient I also in this in this video I want to demonstrate a technique which can be useful for computing the limit of a recursive sequence Because in this context recursive sequences might not have a natural extension to To a continuous or differentiable function and this is going to be based upon what's known as the monotone convergence theorem which basically says the following thing if we have a Bounded function, so there's some value on top that the function never surpasses and there's some value on the bottom that Again, the function never surpasses. We have some maximum value capital M and some minimum value little m I'm not saying the function actually touches these values But we know that it's bounded between these values it never gets above capital M It never gets below little m and also if the sequence is monotonic meaning it's it's always increasing Then the monotone convergence theorem says that the sequence must converge and the proof is basically the following illustration If you're bounded between these two values and your function is increasing It can go up up up up up up up up right, but there's sort of a limit, right? You can't just keep on going up forever your rate of increase kind of has to slow down with time And at some point you're going to taper off at some value and that value Is going to be the limit here So we get some living value so if the function is bounded and monotonic then it'll always always always be convergent So let's see an example why that might be useful. Let's investigate the sequence a sub in that's determined by the recursive Relationship that the base case is a one equals two So that's where it's going to start off and then we have the relationship that the next term of the sequence a n plus One will equal one half a n plus six whenever n is greater than equal to two So if you look at the first couple terms of the sequence, well, let's see those We'll build a chart here. So we have n and a n so one two three four five six That's oftentimes enough to see the pattern here. So it says it starts off at two So if we look at a two, we're gonna plug two into this machine right here You're gonna get two plus two plus six, which is an eight divide that by two You're gonna end up with a four Which is the next term in our sequence Record that here to find a three. We're gonna plug four right here You get four plus six, which is equal to 10 and then 10 over two is going to give you five Which we put in as the next term in our sequence You get a five right there The next one if you plug in five you're gonna get five plus six, which is 11 divided by two gives you 5.5 Uh, which we record here for a four 5.5 the next one I'm gonna be a little bit more careless on the The calculation here if you do a five, you're gonna get 5.5 plus six divided by two That's going to give you 5.75 Then if you take 5.75 plus six, uh, that gives you 11.75 divide that by two You're gonna get 5.875 like so And keep on going with this. We'll do it just a few more terms if you do the seventh term Uh, that would give you 5.9 375 if you do the eighth term that's going to give you 5.9 6 8 7 5 and then lastly If you do the ninth term that would give you 5.9 8 4 3 7 5 So we can see that the initial term If we look at these first couple terms here, we can see that the term appears to be increasing It seems to be an increasing sequence and it seems like it's increasing toward The number six so we actually have sort of a conjecture that maybe Six is the limit of this sequence But how do we actually prove that because we've seen examples in in this course and also in In catechus one where a numerical estimate although helpful can be deceptive, right? How do we guarantee that the limit is in fact six? And so what we're going to do is we're going to use a technique that's referred to as mathematical induction It's a way of making sense out of patterns that are given by these limits or by these sequences Particularly how can we find the limit of this thing mathematical induction? So induction is a method we can use to prove things about sequences. It has three important parts to it So first there has to be some type of base case, right? We have to show that at some point The statement is true, right? So if we want to show that the sequence is increasing We have to be like, okay it increased it increased at some point And so for this one we could say something like a one Is smaller than a two notice here that a one What a one was in fact two, which is less than four So this is a base case that it starts off increasing great Then the next step But the next step here is what's often referred to as the inductive hypothesis The inductive hypothesis What we're going to do here is we're going to assume We're going to assume it holds assume the statement holds For a n So in that context what we mean here is that we're going to assume All right, so we did our base case for this. So we're trying to show that the sequence is increasing here So we showed the base case and so now what we're going to do is we're going to assume Assume that a n minus one is greater than a n and because the idea is if we if we showed One is less than two and then two is less than three and then three is less than four Four is less than five assume. We've got up to this point that a minus one. It will be less than a n okay And so then the third ingredient of our induction argument is We then have to establish the next case This is sometimes referred to as the inductive step We have to then thirdly we use the inductive hypothesis To show the statement holds For a n plus one the next term in the sequence. And so what we have to do is we have to then show That a n is less than a n plus one in our sequence And so we and we do this using the assumption that we had from before And so before we proceed in this in this I kind of want to give an analogy that imagine we're setting up a bunch of dominoes, right? We have a bunch of dominoes In a sequence like the so And maybe see something like this on youtube or something In order for a domino sequence to fall over and be some awesome display There's two ingredients basically first someone has to push the first domino over that's our base case from before But then also the dominoes have to be sufficiently close together so that when they fall They will knock the next one over And so that's what we're trying to do with this step right here That if we push this domino over it'll knock down the the next one in our sequence So for our sequence here, how do we show that it's increasing? Remember our recursive relationship Looks like a n plus one Is equal to one half A n plus six so we're gonna use this we're gonna use this statement to try to prove this and so we're gonna assume remember We're gonna assume That a n minus one is less than a n great So what we do is we're gonna start with this statement a n minus one is less than a n And so what we're gonna do is we're gonna add six to both sides This tells us that a n minus one plus six is less than a n plus six Alright, now we're going to times both sides by one half one half times a n minus one plus six is less than one half a n plus six now the right hand side One half a n minus one plus six that's equal to by this formula right here This is equal to a n and then the right hand side one half a n plus six like we see here This is equal to a n plus one and so we've been proven by induction That our sequence is is increasing See that wasn't so bad right we took our assumption and manipulated until we got the statement We wanted to and this proves that our dominoes fall over one by one by one towards infinity If we were to continue in this manner, so our sequence is an increasing sequence. We've now established that it's increasing by by Induction here, so that's an important thing to mention So if we were to switch the question up is our sequence bounded Is it bounded? Well, because it's increasing We know that a n is going to be greater than or equal to a sub zero I should say a one with the first term in our sequence, which is two So this gives us that it's bounded below because if you're if you're increasing you will always be bounded below That's automatic, but is it bounded above? That one's a little bit harder And again, we're going to use we're going to use induction to help us out here So for our the first the base case our can our following statement's going to be the following a one is less Which itself is equal to two is less than six Remember six was the number we think is the limit here And so we can then show that okay the first term of the sequence is bounded above by six The second step is we're going to assume Assume a n is less than or equal to six All right, so that was easy and the third step is we need to show That a n plus one is less than or equal to six And how we do that is typically taking the assumption we have before and manipulate it remember Kind of like we did it before a n is less than or equal to six. So let's add six to both sides Six plus six that's equals to 12 Times both sides by one half You get one half a n plus six Is less than or equal to 12 over two which is equal to six, but then the right hand the left hand side, excuse me This is just a n plus one And so this then establishes what we are looking for a n plus one is less than or equal to Six and so bobs your uncle there We've now shown that this function is in fact, it's bounded above it's bounded below Therefore this function the sequence is bounded And so why is this important remember how we started this video with the monotone convergence theorem by the monotone convergence theorem We see that the sequence a sub n Is convergent We see that the sequence is convergent that is to say that is a n has a limit Now it has a limit and we suspect that that limit Is six we our claim here is that still the limit is equal to six Now be aware that the monotone convergence theorem does not tell us what the limit is The monotone convergence theorem tells us that a limit exists And because the limit exists we still have to compute what it is now our claim is six And it's a pretty good claim and I want to show you What how how we actually prove it now so important observations the following following if we take the limit As in approaches infinity of a sub n this is equal to some limit value l This will also be the limit as in approaches infinity of a n plus one That is if we just start one term later in the sequence their end destination is still going to be the same thing And so we utilize that in the following calculation here Take the recursive sequence the recursive relation a n plus one is equal to one half A n plus six we are then going to take the limit as n goes to infinity on both sides So take the limit as n goes to infinity on the left and on the right Well on the left hand side it'll converge towards l whatever that limit is And by limit laws on the on the right hand side, we're going to get one half times the limit As n goes to infinity of a sub n and we're going to add six to that Which again the right hand side is going to become one half l plus six and so now we just want to solve this equation for l times both sides of the equation by two How we do this to cancel the one half on the left hand side a right hand side We get two l equals l plus six We're going to subtract l from both sides And then we can see exactly what we're looking for that this limit is equal to six So it's exactly what we claimed it to be Now when you look at this we look at this example here one might be Tempted might be very tempted to add to to to wonder Why did we have to go through all this process? I get how we took the limit of both sides on the recursive relationship And we were able to solve for the limit and get six and so that actually justifies why the limit was six along the way But one has to be very careful because this step right here Uh, if one's not careful could be done erroneously because when you're doing this step you take the limit of both sides You're assuming You're assuming That a sub n is convergent You're assuming a sub n is convergent Otherwise this calculation would be completely meaningless if the sequence wasn't convergent at all And so then assuming it's convergent you would drive this the limit has to be a six But what if the sequence is divergent if the sequence is divergent this this same equation would lead to six But the limit's not six if the sequence is divergent because the sequence Is divergent means the limit doesn't exist So this calculation is only valid if we know the sequence is convergent And the problem happens that there can be divergent sequences which are recursive That if you apply this idea of taking the limits of both sides You will end up with a number, but that's not the limit because it's divergent We can only do this calculation if we know the sequence is divergent And that's why the monotone convergence theorem was necessary And I wanted to mention this because there's a lot of kooky videos online on youtube for which people are Proving bizarre things like pi is equal to four or the harmonic series is equal to what is it supposed to equal to like Two thirds or negative three halves or something absurd like that. How are they getting How are they getting these completely false statements like the harmonic series, which is divergent? How could it possibly equal to a number it doesn't right the issue is they're they're using techniques for convergence theories on Divergent sequences and thus getting these erroneous any ridiculous statements then people are like, oh, it's magic No, it's not magic. It's just incorrect mathematics So in order so finding the limit of a recursive sequence is pretty easy because you just have to do this calculation But be aware this technique only works if the sequence is convergent If the sequence was divergent, we potentially could find a limit that is not correct So be cautious about those