What's the convergence of this series?

Due to the fact that:

  • $\sin(1/n) \sim (1/n)$
  • $\sqrt{n+1} \sim \sqrt{n}$
  • $\log(1/n + 1) \sim e^{1/n}$

I think my general term is equivalent to $e^{1/n}/n^{1/2}$ but i can't do a useful comparison test with this result...

  • $\begingroup$ The last equivalence is certainly false. $log (1+\frac 1 n ) \to 0$ but $e^{\frac 1 n} \to 1$. $\endgroup$ – Kabo Murphy Jun 26 at 9:04
  • $\begingroup$ $\log\left(1+\frac1n\right)\le\frac1n$ and $\sin\left(\frac1n\right)\le\frac1n$ $\endgroup$ – robjohn Jun 26 at 9:46

$\sin (\frac 1 n)$ behaves like $\frac 1 n$ and $\log(1+\frac 1 n)$ also behaves like $\frac 1 n$. So compare with $\sum \frac 1 {n^{1.5}}$ and conclude that the series is absolutely convergent.


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