Chapter 10
Polynomial and Series Representations of Functions





10.6 Convergence of Series

The Divergence Test in the preceding section gives us an easy way to decide that some series do not converge. However, it tells us nothing about convergence or divergence of series whose terms approach zero. In this section we develop several ways to tell whether or not a given series converges. The first of these arises naturally from the examples in the preceding section. It is also the easiest convergence test to apply, but its scope is rather limited. From there we move on to tests that are more robust — and also more difficult to carry out. With each convergence test we also will acquire an error estimate, that is, a way to tell how close any given partial sum is to the total sum. By the end of this section we will have resolved most of the convergence issues for Taylor series representations of important functions.

10.6.1 The Alternating Series Test

Your study of the harmonic series in the preceding section shows that you have to be careful not to jump to conclusions about convergence of series. There is no hope of convergence unless the terms approach zero — the Divergence Test — but "terms approaching zero" is not enough to ensure convergence. In particular, the harmonic series has terms approaching zero, but the sum fails to converge. On the other hand, we can abstract from the examples of the alternating harmonic series and the Leibniz series (both also studied in the preceding section) a general statement about convergence that applies to any alternating series in which the absolute values of the terms steadily decrease to zero.

The Alternating Series Test   Suppose a series has the form

a 1 - a 2 + a 3 - ⋅⋅⋅ + ( - 1 ) k + 1 ⁢ a k + ⋅⋅⋅ ,

where
  • each `a_k` is positive,
  • each `a_k` is larger than `a_(k+1)` , and
  • `lim_(k rarr oo) a_k=0`.
Then
  1. the series converges to some number `S`, and
  2. the error `text[|] S-s_n text[|]` after summing `n` terms is less than `a_(n+1)`.

Activity 1

Use the arguments you have already developed in Activities 3 and 4 of the preceding section to explain why the statement of the Alternating Series Test is true.

Comment 1Comment on Activity 1

Be careful not to read too much into the Alternating Series Test. In one sense, it says that any alternating series that looks like it converges probably does — and it's pretty easy to tell how close any partial sum is to the total sum. But the test certainly doesn't say that all alternating series converge. In fact, it doesn't even say that all alternating series whose terms approach zero must converge. The requirement that absolute values of the terms actually decrease is important. Exceptional cases — in which the absolute values approach zero but not in a strictly decreasing way — are not very important. If you are curious about such things, see Exercise 8.

In order to decide that a given series converges, it's enough that its terms eventually satisfy the conditions of a convergence test such as the Alternating Series Test. That is, we only need to consider all the terms from some point on, a tail of the series. If the tail is a convergent series, then adding in a finite number of terms at the beginning won't affect convergence — although those terms do affect the sum. In a later section we will use the Alternating Series Test to demonstrate convergence of some important series whose tails have the form described here.

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