convergence/divergence of $\sum \left(n\sin\left(\frac{1}n\right)\right)^n$

Ratio and root test both are inconclusive (makes sense since this is after i found the ratio of a power series). i tried some things but it always comes down to $1^\infty$

any tips?


Hint: for large $n$s, $n\sin\frac{1}{n}$ behaves like $1-\frac{1}{6n^2}$, and $\left(1-\frac{1}{6n^2}\right)^n$ is $1-O\left(\frac{1}{n}\right)$, hence the general term is not infinitesimal and the series cannot be convergent.

  • 3
    $\begingroup$ It's not really a hint if it provides the answer, is it? ;) $\endgroup$ – AlexR Oct 3 '15 at 14:42
  • $\begingroup$ @jackd'aurizio Congratulations on reaching 100,000 points!! Well done. $\endgroup$ – Mark Viola Oct 3 '15 at 14:54
  • $\begingroup$ @Dr.MV: thank you so much, I appreciate it. $\endgroup$ – Jack D'Aurizio Oct 3 '15 at 14:56
  • $\begingroup$ definitely an extensive "tip" but helpful all the same $\endgroup$ – Thristle Oct 3 '15 at 23:09

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.