Abel Convergence Test

Consider the series . Suppose that
  1. the partial sums S N = form a bounded sequence
  2. the sequence is decreasing
  3. lim b n = 0
Then the series converges.
Context Context
This test is rather sophisticated. Its main application is to prove the Alternating Series test, but one can sometimes use it for other series as well, if the more obvious tests do not work.

Examples 4.2.18:
 
  • Does the sum converge or diverge ?
  • Does the series converge or diverge ?
Proof:

First, we need a lemma, called the Summation by Parts Lemma:

Lemma: Summation by Parts

Consider the two sequences and . Let S N = be the n-th partial sum. Then for any 0 m n we have:

Proof Proof
Assuming this lemma is proved, we will use it as follows for Abel's Test:

First, let's assume that the partial sums S N are bounded by, say, K. Next, since the sequence converges to zero, we can choose an integer N such that | b n | < / 2K. Using the Summation by Parts lemma, we then have:


     
But the sequence is decreasing to zero, so in particular all terms must be positive and all absolute values inside the summation above are superfluous. But then the sum is a telescoping sum. All that remains is the first and last term, and we have:
But by our choice of N this is less than if we choose n and m larger than the predetermined N. This proves Abel's Test.

What remains to do is the proof of the lemma, which can be found here.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 26, 2007