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?

As far as I can tell, the series is convergent at the origin only. Otherwise (for small $x$) $\log (1+x^2/n)$ is bounded below and above by constant times $x^2/n,$ so diverges by comparison with harmonic series, and for large $x$ the terms are larger than they are for small $x,$ so you are even worse off.
If you are perverse, you can also use the integral test. Integrating the function (with respect to $n$) gives:
$$x^2 \log \left(n+x^2\right)+n \log \left(\frac{n+x^2}{n}\right).$$
Both terms are positive, and the first term clearly blows up.
