Convergence with ln

Likoli

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Nov 1, 2011
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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
ln(1+eax)\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 !
 
I have to study the convergence, depending on the value of a, of the series of
ln(1+eax)\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?
 
I mean studying the convergence of
k=0ln(1+eak)\displaystyle \sum_{k=0}^\infty ln(1+e^{-ak})

Excuse me for not having been clear
 
Last edited:
I mean studying the convergence of
k=0ln(1+eak)\displaystyle \sum_{k=0}^\infty ln(1+e^{-ak})
Clearly it diverges if a0\displaystyle a\le 0, right?

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