0
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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 $0$: $$\frac{\log(k+1)-\log k}{\arctan\dfrac2k}\sim_\infty\dfrac{\dfrac 1k}{\,\dfrac 2k\,}=\frac12$$

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.