# Convergence of $\sum_{n=1}^\infty \sqrt{n + 1} \log(1/n + 1) \sin(1/n)$

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...

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