Test the convergence of the series $\sum_{k=1}^\infty \frac{\log (k+1)-\log k}{\arctan (2/k)}$.

How can I test the convergence of this series using comparison test? I'm out of ideas and I would greatly appreciate any suggestions or solutions.


Use equivalents:

  • $\log(k+1)-\log k=\log\Bigl(1+\dfrac1k\Bigr)\sim_\infty\dfrac 1k$,
  • $\arctan\dfrac2k\sim_\infty\dfrac 2k$,

hence it diverges trivially since the general term doesn't even tend to $0$: $$\frac{\log(k+1)-\log k}{\arctan\dfrac2k}\sim_\infty\dfrac{\dfrac 1k}{\,\dfrac 2k\,}=\frac12$$

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