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Current time:0:00Total duration:8:54

so let's say s is the value that this infinite series converges to we're going to assume that this series actually converges and the the definition of this series each term is going to be a function of n we're going to assume that this is the same type of series that we looked at when we looked at the integral test or namely that this function is a continuous positive decreasing function over the interval that we care about so it is continuous it is positive and it is decreasing this isn't saying continuous plus decreasing it's let me just write it that way continuous positive positive and it is decreasing so it could be a function that looks something like this now my goal of this video is to put is to see if we can estimate a range around s and this is going to be very useful because we've seen some infinite series where we are able to configure out what exactly does it converge to but you could imagine there are many many many more where we're not going to be able to figure out exactly what it converges to and instead we're going to have to do it using a computer or by hand and in those cases it's good to know how good our estimate is and we also want as good of an estimate as possible with as little computation as possible so let's think about how we can do that well the way to tackle it you could imagine is let's split this up this infinite sum let's split it up into the sum of a finite sum so let's say the first K terms so N equals 1 to K of F of n so this is very computable if K is low enough you could probably even and if F is a simple enough function you could probably do this by hand but you could definitely do this with a computer and then it's going to be that plus plus an infinite plus another infinite series but now you're going to pick up at K plus 1 the K plus 1 term and you're going to go to infinity of f of n so if we could put some bounds on this then that'll allow us to put some bounds it'll allow us to put some bounds on this right over because this is just the sum of the partial sum of the partial sum of the first K terms and the remainder that we get after to get us to the actual value so you can see what kind of what's left over after we take that partial sum and this will be this is easier to write for me than this right over here so the key is can we come can we come up with some bounds for this and to do that I'm going to go to this graph and use some of the same arguments we use or the same conceptual ideas we used for the integration test if we there's two ways to conceptual there's two ways to conceptualize what the sum represents relative to this graph as we'll see it can represent an over estimate of the area between some x-value and infinity and it could represent the under estimate of a different region so let's look at that so let's first think about let's first think about the under estimate let's think about the under estimate so if this right over here let's say that this right over here is K actually let me do it in a color that you can it's let me do it in that we do in a yellow color so let's say that this right over here is K this is k plus 1 this is k plus that's and behind that's let's do this is k plus 2 k plus 2 k plus 3 on and on and on so one way to conceptualize this sum right over here is it could be the sum of the following rectangles so the first term is the this area the area of this first rectangle because this area is this area is height or this rectangles height is f of k plus 1 is f of k plus 1 and its width is 1 so f of k plus 1 times 1 its areas is going to be f of k plus 1 which is exactly this first term right everyone n is k plus 1 and then the second term by the same argument could represent the area of this rectangle the third term could represent the area of that rectangle and we could just keep going on and on and on and so what are the what is the sums of these areas of these rectangles so the sum of these terms are presenting well you could view it as an estimate you could view it as an estimate of the area under the curve between x equals k and x equals infinity but it's going to be an under estimate notice the these are all completely contained in that area so one way to think about it is that our r sub k our sub K is less than or equal to it's under estimate for the area between X equals K between X equals K and infinity of f of X f of X DX so that essentially puts an upper bound that puts an upper bound on us and this is already interesting because now we can already say if s so we know that s s is equal to this now if this is less than this thing right over here we can say that s is going to be less than or equal to R our partial sum plus plus this thing plus the improper integral from K to infinity of f of X DX notice if this is equal to this and now since this is less than this this must be less than what we have on the right hand side so just like that if we're able to compute these two things and we are often able to compute these two things we're able to put an upper bound on our actual sum now what about placing a lower bound on it well we could conceptualize the same sum the same R sub K instead instead we can conceptualize it this way where the first term here represents not this rectangle but it represents this rectangle notice it has the same height but it's just shifted over one to the right the second term represents this rectangle the third term represents this rectangle why does that make sense well the area of this first rectangle is is going to be its height which is f of K plus 1 f of K plus 1 times its width which is this one so it's just going to be f of K plus 1 so this is the area here is the first term the area here is the second term area here's the third term area here is a fourth term one way to think about it we just shifted all of those yellow rectangles ones one to the right but now this is approximating a different region this is approximating the area under the curve not from K to infinity but from K plus 1 to infinity and instead of being an underestimate it's an overestimate now the curve is is contained within the rectangles so we could say that R sub K in this context when we conceptualize this way is going to be greater than or equal to the improper integral not from K but from K plus 1 from K plus 1 to infinity f of X DX and what does that allow us to do well this place is a lower bound on this which will also place the lower bound on this if this is greater than that then this is going to be greater than this replaced with the improper integral so we can write that so s is going to be greater than or equal to S sub K plus plus the improper integral from K plus 1 to infinity f of X f of X DX now you might be saying hey Sal this looks all crazy you have all this abstract notation here you got to introduce an integral sign this seems really daunting but as we'll see in the next few videos these are actually sometimes fairly straightforward to compute this can be very straightforward to compute if RK is not too large if it's even large a computer can do it and then these we can actually often compute sometimes numerically but even more frequently and that kind of defeats the purpose but we can also compute them using our analytic tools do it using it using I guess you could say the the the power of calculus and so what this this allows us to do is put a pretty neat band around what our actual value that we converge to is and as we'll see the higher our K the better and estimate we get and the tighter arrange of our confidence for that estimate and another way to write these two inequalities is to write a compound inequality that S is going to be less than or equal to this business it's going to be less than or equal to this business so copy and paste and it's going to be and s is going to be greater than or equal to this business so you can say this business is going to be less than or equal to s so let me just copy and paste that so copy and pay ups that's not what I wanted to do so copy and paste now we could write it just like that and so the next series of videos will actually apply this and we'll actually we'll see that it's pretty straightforward it looks a little daunting right now you