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

1. for all

2.

3. for all

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

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

.

Then one can combine the terms in the parenthesis and then do a limit comparison to the series to see the series diverges.