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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?

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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.

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  • 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

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