The following seems to be a question related to standard calculus, but I am not quite sure where to look for an answer.

Suppose $f,g:\mathbb{N} \to \mathbb{C}$ are such that the have the same asymptotical behaviour, i.e. $f(n)/g(n) \to 1$ as $n \to \infty$. Of course, suppose that one of the sums $\sum_{n=0}^\infty f(n)$ and $\sum_{n=0}^\infty g(n)$ converges absolutely, then so does the other. This can be proven by a standard estimate. However this standard estimate fails if we do not have absolute convergence. I do not see how to prove convergence of one of sums implies the convergence of the other. I feel that it may be actually false.

So the first question is:

$1$. Is it true that one series converges iff the other does?

If this is not the case, however, in the problem I am studying, I want to prove convergence for both series. For my application in mind, you may assume that $f(n)/g(n)$ is always in $\mathbb{R}$. So the second question is

$2$. Under which additional conditions (which do not! imply absolute convergence) can we deduce both series have the same behaviour. Are there books treating such topics?

`$\delta_n$`

s or their sizes --- if`$\sum \delta_n$`

converges absolutely, for example, then you're golden, and this requires only that the approximation $f(n) \approx g(n)$ gets sufficiently better as $n\to \infty$. Emerton: you should leave your comment as an answer. $\endgroup$