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?
> 
> 
> 



Non-absolute convergence of series with asymtotically  equal coefficients
Following Theo Johnson--Freyd's suggestion, I am making my above comment an answer:


Just subtracting one series from the other, it seems that you need $\sum\_{n=0}^{\infty} (f(n)−g(n))$ to converge. Writing $$f(n)/g(n)=1+\delta\_n,$$ $$\text{so that } \qquad \qquad\sum\_{n=0}^{\infty} (f(n)−g(n))=\sum\_{n=0}^{\infty} \delta\_n g(n),$$ you see need control over the signs of the $\delta\_n$, or (as Theo notes in his comment, on their
rate of growth).


E.g. if they are all of the same sign, you are okay, while if the sign of $\delta\_n$ is always the same as, or always opposite to, that of $\delta\_n$, then you could be in bad shape. (This is what goes wrong in Xandi Tuni's example.) 


As Theo notes in his comment, you are also okay if $\sum\_{n = 0}^{\infty} \delta\_n$
converges absolutely. Whether this
applies in your case will depend on how closely $f$ and $g$ approximate one another.
(This is illustrated by the example in Julian Aguirre's answer.)


