It is well known that if series meets the following conditions:
1. for all
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.