# Studying the convergence of the series $\sum_{n=1}^\infty\sin\frac1{n}\log\left(1+\sin\frac1{n}\right)$

Study the convergence of the series

$$\sum_{n=1}^\infty\sin\frac1{n}\log\left(1+\sin\frac1{n}\right)$$

This is what I came up with

$$\lim_{x\to \infty}\frac{\sin\frac1{n}\log\left(1+\sin\frac1{n}\right)} {\sin^2\frac1{n}}= 1$$ This implies that

$$\sin\frac1{n}\log\left(1+\sin\frac1{n}\right) \sim {\sin^2\frac1{n}}$$

using the inequality $\sin{x}\lt x$ $\left(0\le x \lt \pi\right)$

$${\sin^2\frac1{n}} \lt \frac1{n^2}$$

Since $\sum_{n=1}^\infty\frac1{n^2}$ converges so does $\sum_{n=1}^\infty\sin^2\frac1{n}$ this implies the convergence of $$\sum_{n=1}^\infty\sin\frac1{n}\log\left(1+\sin\frac1{n}\right)$$

Is this right?

• You are correct! – Parcly Taxel Jul 25 '18 at 16:51
• Thank you. I was not very sure. – J.Dane Jul 25 '18 at 16:55