Determining convergence of series of functions with natural logarithm

Question

I have to determine the convergence of the series of functions of $\sum f_n$, $f_n:\mathbf R \rightarrow \mathbf R$, defined by: $$f_n = \biggl(\frac{\ln(1+n^2x^2)}{n}\biggr)^n$$

I do not know how to proceed, any help will be appreciated.

Answer 1

For $x=0$, it converges.

If $x\ne 0$, it is a positive terms series.

The root test gives $$\lim_{\infty}\frac {\ln (1+nx^2)}{n}=0$$ since $$\ln (1+nx^2)\le \sqrt {1+nx^2}$$ and $$\sqrt {1+nx^2}=\sqrt {n}\sqrt {x^2+\frac {1}{n}} $$

Thus, it converges for all reals.