Convergence with ln

[Likoli]

Hello everyone !

Please excuse every mistake that I may make in English, it's not my first language.

I have a little problem : I have to study the convergence, depending on the value of a, of the series of
\(\displaystyle ln(1+e^{-ax})\)

Obviously, it is divergent when a is negative or equal to 0. My problem is that I can't remember how to solve this when a is positive, it's quite far in my memory !

Thanks for the help you may provide !

 

[pka]

I have to study the convergence, depending on the value of a, of the series of
\(\displaystyle ln(1+e^{-ax})\)

How are you using the word 'series' here?
Do you mean sequence, limit or series representation of that function?
What is the full and exact statement of the question?

 

[Likoli]

I mean studying the convergence of
\(\displaystyle \sum_{k=0}^\infty ln(1+e^{-ak})\)

Excuse me for not having been clear

 

[pka]

I mean studying the convergence of
\(\displaystyle \sum_{k=0}^\infty ln(1+e^{-ak})\)

Clearly it diverges if \(\displaystyle a\le 0\), right?

Suppose \(\displaystyle a>0\) then \(\displaystyle \ln\left(1+e^{-ak}\right)=\ln\left(e^{ak}+1\right)-ak\). SO?