A more general criterion of convergence is that and converges...
Kind regards
A part of my course complex-analysis we deal with infinite products. And as a homework assignment I got the following exercise.
If for all , then prove that converges iff
I'm used to not so easy homework-assignments, so that's why I feel I'm missing something here.
My answer would be:
if we have since for all . Hence and thus we obtain
If we have
So am I missing something?
Go back to the definitions. To say that converges means that exists (and is nonzero). To say that converges means that exists. Also, if for all n then is a series of positive terms, so it converges iff its sum is finite.
Now use the fact that the logarithm function and its inverse are continuous to argue that converges iff converges, as .