Here's a thought...
Let's take two series, call them A and B.
Let's say each term in A is less than or equal to the corresponding term in B.
Logic would suggest that, if B is convergent, then A is convergent too.
So how about this...
Let B be the geometric series 1 + 2/3 + 4/9 + 8/27 + ... + (2/3)^N + ...
This series converges to 3.
Let A be the series 1 + 1/2 + 1/3 + 1/4 + ... + (1/N) + ...
This series is not convergent.
However, each term of A is less than or equal to the corresponding term in B, i.e.
1 <= 1 1/2 <= 2/3 1/3 <= 4/9 1/4 <= 8/27 ... 1/N <= (2/3)^N ...
If B is convergent, doesn't this mean that A should be convergent too?
Hmmmm...
M.
