Test convergence of $\sum_{n=1}^{\infty} \dfrac {1}{\log(e^n+e^{-n})}$

Attempt: I have tried the integral test, the comparison test ( for which I couldn't find a suitable comparator).

However, I haven't been able to proceed much. Could anyone please give me a direction on how to proceed ahead with this problem.

Thank you very much for your help in this regard.


2 Answers 2


$e^{n}+e^{-n}< 2e^{n}$, hence $$\frac{1}{\log(e^n+e^{-n})}>\frac{1}{n+\log 2}$$ and the series is divergent by direct comparison with the harmonic series.

  • $\begingroup$ Is someone having fun in distributing random downvotes? $\endgroup$ Jan 25, 2015 at 17:56
  • 1
    $\begingroup$ Thank you for your answer :) . Someone's just randomly downvoted all the answers and the question too. $\endgroup$
    – MathMan
    Jan 25, 2015 at 17:58

Hint: compare with the series $\sum_n \frac{1}{n}$.

  • $\begingroup$ You are welcome! $\endgroup$ Jan 25, 2015 at 18:07

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