In the previous video, I presented the Alternating Series Theorem (or Test),
which allows us to prove
that a large family of alternating series are convergent.
I assume you have watched that video first,
otherwise this one won't make sense.
In this video, I want to take things one step further,
and show how we can not only conclude they are convergent,
but easily estimate the value of their sum.
I'm going to do this with an example. Here is the exercise:
I'm looking at the sum from n = 1 to ∞, of (-1)ⁿ/ n⁴,
and I want to estimate the value of this sum, with an error smaller than 0.001.
Before I do that, first, I need to show that this series is convergent,
otherwise it doesn't make sense to calculate it's value.
And yes it is convergent,
because it satisfies the hypotheses of the Alternating Series Theorem.
As a quick reminder, this is the theorem I proved in the previous video.
If I have a series of this form:
(-1)ⁿ times something positive, or sum of (-1)ⁿ⁺¹ times something positive
(that means it's alternating), with two extra hypotheses
(that the sequence of things I'm adding or subtracting
is decreasing, and has limit 0),
then necessarily, the series is convergent.
I went quickly through this.
This is the theorem I proved in the previous video.
If it feels unfamiliar, please watch that other video first.
We can use that theorem in this case for the example I'm looking at
(sum from n = 1 to ∞, of (-1)ⁿ/n⁴)
because it's alternating, and 1/n⁴ is decreasing and has limit 0,
so the series is convergent.
So at least the question makes sense.
This series (this infinite sum) is a number.
Now what is that number? That's what I want to know.
By definition, an infinite sum is defined as a limit of finite sums
(as a limit of partial sums Sₖ).
This suggests that I could use Sₖ as an estimate.
I remind you Sₖ is the sum of the same terms,
where I stop at k, instead of continuing going to ∞.
This is a priori a good idea, because a finite sum is something I can calculate.
I'll probably have to take a large value of k.
If S is the limit of the sequence Sₖ,
Sₖ for a very large value of k, is probably a good estimate for S.
But I want to know exactly which value of k do I need to use?
I have, in this case, a very precise criterion:
I want the error to be smaller than 0.001.
Well, the error of the estimation
will be the distance between the actual value S, and the estimate Sₖ.
So how do I choose a value of k that makes the difference...
...between S and Sₖ in absolute value, smaller than 0.001?
In principle, this idea of using a partial sum to estimate an infinite sum,
is something we could use for any convergent series.
After all, a convergent series is always defined as the limit of finite sums.
The advantage of this type of alternating series,
is that we are going to have one easy quick way,
to figure out which value of k to use.
Here is why.
This is the picture I drew in the previous video,
that shows the partial sums of a series like this.
I start at 0, I add something, then I subtract something smaller,
then I add something smaller, subtract something smaller, and so on...
This series is convergent,
and the value of the series (the value of the infinite sum),
is somewhere here. It's in the middle.
It's > all the even partial sums, and < all the odd partial sums.
Now I am interested in the error of an estimation by a partial sum.
So for example, if I stop at this partial sum,
or if I stop at this partial sum, what is the error I make?
Well, let's see. If I stop at this partial sum,
the error I am making is the distance between the actual value of the series,
and the value of the partial sum;
and that error is smaller than the first term I am NOT adding.
Or if instead I stop at this partial sum,
then I want to see the error I'm making
(the distance between this partial sum and the actual value of the series);
and that error is smaller than the first term I am NOT subtracting.
So in all cases, the error I make
is smaller than the first term I am not adding or subtracting,
and it should be clear that this only applies to this specific type of series.
I am going to call this a theorem.
In fact, it is the second part of the Alternating Series Theorem.
IF I have a series alternating like before,
(and I'm going to be lazy and just say:
"that satisfies the same three hypotheses as before"
(I'm not going to copy them again)),
THEN not only do I know that it is convergent,
I have a way to estimate or bound the error I make,
when I approximate it with a partial sum.
The distance between the actual value of the infinite sum,
and the value of the kth partial sum,
is smaller than the first term, that I am not adding or subtracting.
That's it. It's simple, and it's convenient.
Let's go back and finish that exercise. This was the question.
I have this infinite sum, and I am estimating it with the kth partial sum.
I was concerned about the error of the estimation.
I now know it is smaller than the first term I am NOT including.
Since I am cutting the infinite sum at the kth term,
the first term I am NOT including is the (k + 1)st, or 1/(k + 1)⁴.
So for my purpose, what I need to do is choose a value of k
such that 1/(k + 1)⁴ is < 0.001, that's it! The rest is algebra.
At this moment, I invite you to pause the video,
and just do a bit of algebra and solve this inequality.
Which integer k satisfies this?
If you go through that, you will get that k = 5 it's the smallest one that works,
and to make my life simpler, I'll take the smallest one that works.
So let's finish this.
My estimate is going to be S₅,
which is the sum of the first 5 terms in the series, that's it!
I can compute this easily: the first few decimal digits are -0.94753.
I can actually compute the exact value of this infinite sum with other methods.
There's a way to do it, although it's much more complicated
(and it's too advanced for what we have at the moment),
so I won't explain how to do it;
but I can compute the exact value of the series,
and the first few digits are -0.94703, so as promised, this worked.
With this simple estimate (just 5 terms), I got an error that was < 0.001.
In general, for other kinds of series (for Taylor Series),
there are a few theorems that give us a way to bound the error that we make,
when we estimate an infinite sum with a partial sum.
But none of them are as simple as this calculation that we can do
for alternating series.