calculus - convergence of $sum_{n=1}^{infty} frac {1}{log(e^n+e^{-n})}$?

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.

Answer

$e^{n}+e^{-n}< 2e^{n}$, 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.