Does this series converge or diverge? $\sum_{n=1}^\infty\ln\left(1+\frac1{n^2}\right)$

Hint: Recall that $\ln(1+x)\sim x$ for $x\to 0$, and use the fact that $\sum_{n=1}^\infty\frac1{n^2}$ is convergent.

By the Mean Value Theorem
$$\ln(1+x) = x \cdot\frac{ 1}{1+cx}\leq x $$
where $0 < cx < x$. Hence $0\leq\ln(1 + 1/n^2)\leq 1/n^2$ for each $n\geq 1$. Then $$\sum_{n=1}^{+\infty}\ln\left(1+\frac{1}{n^2}\right)\leq \sum_{n=1}^{+\infty} \frac{1}{n^2}=\frac{\pi^2}{6}.$$