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

- [Voiceover] 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 definition of the
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. Let me just write it that way. Continuous positive and it is decreasing. So it could be a function that
looks something like this. Now, my goal of this video
is to see if we can estimate a range around S. So this is going to be very
useful, because we've seen some infinite series where
we are able to figure out what exactly does it converge to. But you could imagine
there are 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 one to k of f of n. So this is very computable. If k is low enough and if
f 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 an infinite. Plus another infinite series. But now you're going to
pick up at k plus one, the k plus one 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
on this right over here. 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 could see kind of
what's left over after we take that partial sum. And this is easier to
write for me than this right over here. So they key is, 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 used, or the same conceptual ideas we used,
for the integration test. 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 sum,
x value, and infinity. And it could represent
the underestimate of a different region. So let's look at that. So let's first think
about the underestimate. Let's think about the underestimate. 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's... Let me do it in that yellow color. So let's say that this
right over here is k. This is k plus one. Let's do this is k plus two. 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 this
area, the area of this first rectangle. Because this area's height,
or this rectangle's height, is f of k plus one. And its width is one, so
f of k plus one times one. Its area is just going to
be f of k plus one, which is exactly this first term
right over here, where n is k plus one. And then the second term,
by that same argument, could represent the
area of this rectangle. The third term could represent
the area of the rectangle. And we could just keep
going on and on and on. So what are the sums of these
areas of these rectangles? What are the sums of
these terms representing? 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 underestimate. Notice these are all completely
contained in that area. So one way to think about
it is that our R sub k is less than or equal to its
underestimate for the area between x equals k and
infinity of f of x dx. So that essentially puts
an upper bound on us. And this is already
interesting, because now we can already say if S -- so we
know that 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 our partial sum 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 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. This third term represents this rectangle. Why does that make sense? Well, the area of this first
rectangle, this is going to be its height, which is f of
k plus one times its width, which is just one, so it's
going to be f of k plus one. So the area here is the first
term, the area here is the second term, area here is
the third term, the area here is the fourth term. One way to think about it,
we just shifted all of those yellow rectangles 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 one to infinity. And instead of being an
underestimate, it's an overestimate. Now the curve is contained
within the rectangles. So we could say that R sub
k in this context, when we conceptualize it this way,
is going to be greater than or equal to the improper
integral not from k, but from k plus one to infinity f of x dx. And what does that allow us to do? Well, this places a lower
bound on this, which will also place a 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 the improper integral
from k plus one to infinity f of x dx. Now, you might be saying,
"Hey, Sal, this looks all "crazy, you have all this
abstract notation here, you've "introduced 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 our k 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,
using, I guess you could say, the power of calculus. And so what 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 an 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. So copy and paste. And S is going to be greater
than or equal to this business. So we could say this
business is going to be less than or equal to S. So let me just copy and paste that. So copy and paste. Oops, that's not what I wanted to do. So copy and paste. Then we could write it just like that. And so the next series of
videos will actually apply this and we'll see that it's
pretty straightforward. It looks a little daunting right now.