Uniform convergence of functional series

Question

I can't prove ( or refute ) uniform convergence of $\sum\limits _{n=1}^{\infty}\frac{e^{\frac{-x}{n}}\cos nx}{x^{2}+n^{2}x}$ $,\,\,0.01<x<+\infty$

I tried Weierstrass M-test, but failed. Other methods didn't work too. Can't find the soulution to this.

Answer 1

Justify $$ \left|e^{\large -\frac{x}{n}}\cos nx\right|\le e^{\large -\frac{0.01}{n}}\left|\cos nx\right|\le e^{\large -\frac{0.01}{n}}< 1, \quad x \in (0.01,\infty), $$$$ \left|\frac{1}{x^{2}+n^{2}x}\right|< \frac{1}{0.01}\cdot\frac{1}{n^{2}+0.01} < \frac{1}{0.01}\cdot\frac{1}{n^{2}}, \quad x \in (0.01,\infty), $$ then conclude.