Abel's Test

Consider the series

. Suppose that

the partial sums S _N = form a bounded sequence
the sequence is decreasing
lim b _n = 0

Then the series

converges.

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

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:

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. Therefore, 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.

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