# 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.