 Welcome back everyone we want to continue our discussion about sequences which is a discrete functions functions which have gaps inside of their domain. And we saw in the previous video that while computing limits of sequences in many ways is just like computing limits of continuous functions we have seen some examples where because of issues like say in fact or. Well it doesn't have some continuous expansion to x factorial we might need new tools to help us compute limits in some situations. And so for example we don't have derivatives we couldn't use L'Hopital's rule in that previous situation but we can still use derivative like arguments. That is we can still talk about monotonicity about about sequences and so what does it mean for a sequence to be increasing. Well we often so much think about increasing meaning the derivative is positive but increasing really just means that the next term is bigger than the previous term. That's what it means for a sequence to be increasing and what does it mean for a sequence to be decreasing well means the next term is smaller than the previous term. That's what it means for a sequence to be decreasing and so we say that a sequence is monotonic if it's either increasing or decreasing. So we think of monotonic as this gender neutral term to describe an increasing or decreasing sequence. And so this is going to this is related this is going to be like our discrete idea of a derivative a sequence is increasing or decreasing how does one describe this. Well another term that's related to this is the idea of a bounded sequence we say a sequence a sub is bounded above if there exists some number m which is bigger than all the terms in the sequence. Right so m is bigger than everything in the sequence is bounded above similarly we say a sequence is bounded below if there's some little number m that's smaller than all the terms in the sequence. And so this idea being bounded above and bounded below is a discrete analog of the idea of finding an absolute maximum or absolute minimum on the graph. Okay and we say that a sequence is bounded if it's bounded above and below so let's take a look at examples of some of these things. So take the sequence of three over n plus five that you see right here is this sequence decreasing. Well if we want to show this generally speaking take take the nth term in the sequence three over n plus five then look at its that its successor right look at the n plus one term. This would mean I kind of got covered up a little bit there this would mean we're going to replace the n with an n plus one and see what happens there. So if we replace n with n plus one the denominator is going to become n plus six the numerator stays as a three. An important thing to remember when it comes to fractions if you take one over a big number this is actually equal to a smaller number that is to say making the denominator get bigger actually makes the ratio get smaller. So as you went from n plus five to n plus six the numerator stayed the same but the denominator got bigger so that made the whole thing get smaller right here. And as such this is evidence that your sequence is decreasing because the next term got smaller and this is not just true for an individual term this is true for everything right. We took the nth term of our sequence and we've now shown that the nth term is greater than the n plus one term the next term in the sequence and so this demonstrates that our sequence is decreasing. Well is this sequence is this sequence is it bounded right well if your sequence is bounded or I should say if your sequence is decreasing that it kind of has to be bounded above right because if your sequence is bounded or if it's decreasing it's be bounded above by the first term in the sequence. Take for example a zero here the zero term of the sequence you take three over zero plus five that's three fifths every term of the sequence is going to be smaller that because this right here is bigger than a one which is bigger than a two which is bigger than a three etc. Every term is smaller than that so decreasing sequences will always be bounded above but we'll also notice that it's bounded below because each of these each of these terms is is positive so it's greater than zero. And in fact what we see right here is we take the limit as in approaches infinity of our sequence three over n plus five. This limit will equal zero all of the terms in the sequence are positive but they're going to get smaller and smaller getting closer and closer to zero. So this sequence is a decreasing sequence was is bounded bounded above by the first term a sub zero and it's bounded below by its limit we see that a lot. As another example let's show that the sequence a sub n where it's n over n squared plus one this we also want to show this is a decreasing sequence. While in the previous example we were just able to replace n with n plus one and we're good as good as gold there if we try that here a n equals n plus one over n plus one squared plus one. We have a little bit more careful here because we notice in this example the numerator gets bigger and when the numerator gets bigger they typically means that the fraction got bigger but we also saw that the denominator got bigger as well. And so if the numerator and the denominator simultaneously get bigger then that's a little bit harder to see what's going on here like there's a clash of type more powerful did this thing the whole fraction get bigger the whole fraction gets smaller or maybe nothing happened right. So we feel a bit more sophisticated here but because n over n squared plus one because we can extend this to a continuous function. While the sequence does not have a derivative the continuous expansion does for which we can compute the derivative by the usual product rule we get low d high minus high d low square the bottom here we go. If you simplify the numerator we're going to get x squared or x squared plus one minus two x squared over x squared plus one quantity squared combining like terms you end up with one minus x squared over one plus x squared squared. Now I should mention that the denominator here is always going to be positive x one plus x squared is a positive quantity and then when you square something that's real it's always going to be non negative so the denominator is strictly positive it'll have no effect on one that derivative is negative because remember the derivative being negative means that the function is decreasing which is what we're trying to figure out right here. So when does the numerator one minus x squared when is that equal to zero. Well this is a very simple thing this factor one minus x times one plus x right here equals zero and so we get x could be plus or minus one in this situation. Now of course negative ones outside the domain of the sequence it's part of the real function but we have to only take positive inputs right here and so what we see happening is that if we restrict in to be greater than equal to one which is what we do for these sequences here. Then in that situation our function f prime of x is going to be a negative and so this is actually evidence that the sequence is decreasing. So I kind of want to mention here that when we can extend our sequence to a continuous function I should say a differentiable function. Then we can use its first derivative to determine whether it's increasing or decreasing exactly the same way that we do it in calculus one the domain of these things for our sequences or domains can be greater than equal to one right. But and on the previous example we could actually done that as well right we could have done f of x equals three over x plus five and then by just usual rules of derivatives. So we end up with negative three over x plus five squared, in which case we can see that this number is always negative for any choice of acts, other than negative five of course where it's undefined but that's outside the domain. So we could have done a derivative calculation to make this thing very clear that this thing was decreasing that applies rules from calculus one can be used to help us out here. But the point is because these functions are discrete sequences we potentially could use alternative methods from derivatives to determine whether the functions monotonic or whether it's bounded and in some situations like this one they're much easier. But in situations like this one it can get a little bit messy for which we might want to fall back onto the notion of the derivative to help us out.