Does this series converge or diverge? $\sum_{n=1}^\infty\ln\left(1+\frac1{n^2}\right)$ I have a series here, and I'm supposed to determine whether it converges or diverges. I've tried the different tests, but I can't quite get the answer.

$$\sum_{n=1}^\infty\ln\left(1+\frac1{n^2}\right)$$


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  $$\sum_{n=1}^{+\infty}\ln\left(1+\frac{1}{n^2}\right)\leq \sum_{n=1}^{+\infty} \frac{1}{n^2}=\frac{\pi^2}{6}.$$
