4.2. Convergence Tests
Abel Convergence Test
Consider the series . Suppose that
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.
Proof:
- 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:
Lemma: Summation by Parts
Assuming this lemma is proved, we will use it as follows for Abel's Test:
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.