Uniform convergence research

Question

I got a task: research

$$\sum_{n=1}^\infty e^{-n\tan (x)}, 0<x<\frac{\pi}{2}$$

for a uniform convergence. Can anybody help?

Answer 1

Hint. Assume $\alpha>0$. For $\alpha\le x<\dfrac \pi2$, one has $$ 0<e^{-n \tan x}\le e^{-n \tan \alpha}<1 $$ giving, as $N \to \infty$, $$ 0<\sup_{[\alpha,\pi/2)}\left| R_N(x)\right|=\sup_{[\alpha,\pi/2)}\left|\sum_{n=N}^\infty e^{-n \tan x}\right|\le \sum_{n=N}^\infty e^{-n \tan \alpha}\to 0 $$ that is the series is uniformly convergent over each $[\alpha,\pi/2)$.

Can you finish it by proving the convergence is not uniform over $[0,\pi/2)$?