Uniform convergence of the series $\sum_{n\ge 1}\frac{ne^{-nx}}{1+n^2}$

Question

I'm triying to prove the uniform convergence of the series $$\sum_{n\geq 1} \frac{ne^{-nx}}{1+n^2}$$ for $x\in (0,\infty)$, I tried to use the M-test many times, but only can acomplish this for $x\in[a,\infty)$, $a>0$. Can be an error in the exercise?

Answer 1

If the series was uniformly convergent on $(0,+\infty)$ then for each $\varepsilon>0$ there would be an $N$ such that $$ \sum\limits_{n = N}^\infty \frac{ne^{ - nx} }{1 + n^2 } < \varepsilon $$ for all $x>0$. Note however, that for $N\geq 1$ with $x=1/N$, $$ \sum\limits_{n = N}^\infty {\frac{{ne^{ - n\frac{1}{N}} }}{{1 + n^2 }}} > \sum\limits_{n = N}^{2N} {\frac{{ne^{ - n\frac{1}{N}} }}{{1 + n^2 }}} > e^{ - 2} \sum\limits_{n = N}^{2N} {\frac{n}{{1 + n^2 }}} > e^{ - 2} \frac{{2N^2 }}{{1 + (2N)^2 }} > 0.05. $$ Thus, if $\varepsilon<0.05$, there cannot be such a positive $N$.