Abel Convergence Test
- the partial sums S N = form a bounded sequence
- the sequence is decreasing
- lim b n = 0
First, we need a lemma, called the Summation by Parts Lemma:
Consider the two sequences and . Let S N = be the n-th partial sum. Then for any 0 m n we have:
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.