There is a nice result for approximating the remainder of convergent alternating series.
Recall that if is a sequence of positive terms, we say the series and are alternating series, and we have a nice result to test these for convergence.
- eventually, and
then, the alternating series converges.
Note that this test gives us a way to determine that many alternating series must converge, but it does not give us information about their corresponding values. If we want to approximate such series, we must study their remainders.
As usual we must establish that a series converges first before we begin to think about remainders. Once we have established that an alternating series converges, we have the usual decomposition.
As before, is the approximate value of the infinite series and is the error made when using this approximation. While we cannot find an explicit formula for , we have a good way to establish bounds on the error made when approximating by the finite sum . Let’s explore this result pictorially for a general alternating series.
- for every .
- is strictly decreasing immediately; that is for every .
- (not really an assumption since converges).
- The value of the convergent series is the number .
Let’s plot the terms of two sequences : , which consists of positive terms and the sequence of partial sums for the alternating series . We first start with the plot of .
We now plot the terms in the series . Note that when written out, the sum in question is
That is, we alternate between adding and subtracting terms in the sequence .
Note that in the above picture, the term corresponding to the error made when we use to approximate the infinite series is the difference between the actual value of the series, and .
We now note the following. For odd indices , we have (from the picture) the relationship below. By the same logic about the relative sizes of the terms, we have for even that
Now, we can think of the remainder as the difference between and at any stage, or that
Of course, if is negative, we have the same behavior, except the odd partial sums are increasing and the even ones are decreasing. Try it out with an example if you are skeptical! We can now state the general result for approximating alternating series.
- , and
then, we have the following estimate for the remainder.
In order to gain some practice, let’s work an example.
Simple enough! Let’s see if our other typical question presents any additional trouble.
We know that so if we can force our upper bound to be less than , then the size of the remainder can be no larger. Now
is satisfied for Taking the square root of both sides, we see that we need or In other words, when , we have
Finally, let’s consider one more problem.
When is a decreasing sequence of positive term, we will approximate by the finite sum . Generally, adding more and more terms of a convergent series should generally get you closer to the actual sum! Indeed, we have a nice bound for the remainder:
We can use to approximate the series to any degree of accuracy as we want!