Interval of convergence of $\sum \limits_{n=1}^\infty\ln(1+\frac{x^2}{n})$ How to determine the convergence interval of the $\sum \limits_{n=1}^\infty\ln(1+\frac{x^2}{n})$  function series? The required condition is proven, it can be convergent, but where?

Note that $\ln(1 + t)\sim t$ as $t\to 0$.  So
$$\ln(1 + x^2/n) \sim{x^2/n}.$$  
