College Math Teaching

October 29, 2015

The Alternating Series Test: the need for hypothesis

Filed under: calculus, series — Tags: — collegemathteaching @ 9:49 pm

It is well known that if series \sum a_k meets the following conditions:

1. (a_k)(a_{k+1}) < 0 for all k
2. lim_{k \rightarrow \infty} a_k = 0
3. |a_k| > |a_{k+1} | for all k

the series converges. This is the famous “alternating series test”.

I know that I am frequently remiss in discussing what can go wrong if condition 3 is not met.

An example that is useful is 1 - \frac{1}{\sqrt{2}} + \frac{1}{3} - \frac{1}{\sqrt{4}} + ...+\frac{1}{2n-1} - \frac{1}{\sqrt{2n}} .....

Clearly this series meets conditions 1 and 2: the series alternates and the terms approach zero. But the series can be written (carefully) as:

\sum_{k=1}^{\infty} (\frac{1}{2k-1} - \frac{1}{\sqrt{2k}}) .

Then one can combine the terms in the parenthesis and then do a limit comparison to the series \sum_{k=1}^{\infty} \frac{1}{k} to see the series diverges.

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