Uniform convergence of the series $\sum\limits_n \frac{\log(1+n^2x^2)}{n^2}$ on $[0,1]$

Question

Is the series $$\sum_1^\infty \frac{\log(1+n^2x^2)}{n^2}$$ uniformly convergent on $[0,1]$? I tried to use comparison test but unable to bound this by any convergent series. Any suggestion will be very helpful.

Answer 1

Note that $$ \log(1+n^2x^2)\leq \log(1+n^2) $$ for $x\in [0,1]$ by monotonicity of the log. Use Weirstrass M-test to conclude.

edit: To see that the series $$ \sum\frac{\log(1+n^2)}{n^2} $$ converges note that $$ \frac{\log(1+n^2)}{n^2}\sim \frac{\log n}{n^2}<\frac{n^\epsilon}{n^2}=\frac{1}{n^{2-\epsilon}} $$ for any epsilon greater than zero and correspondingly large $n$ (a very useful fact about the logarithm). Probably going to want to select an $\epsilon<1$.