Uniform convergence of $\sum_{n=1}^\infty \frac{1}{z^n+1}$ I've proved that the series $\sum_{n=1}^\infty \frac{1}{z^n+1}$ for $|z|\geq r >1,z\in\Bbb C$ converges uniformly by weierstrass M-test. But I want to show it's not uniformly convergent on $|z|>1$. Could you give any hints?

If it was uniformly convergent for $|z|>1$,
$$
\mathop {\sup }\limits_{|z| > 1} \left|\sum\limits_{n = N}^\infty  {\frac{1}{{z^n  + 1}}} \right|
$$
would tend to zero as $N\to +\infty$. Note however that
$$
\mathop {\sup }\limits_{|z| > 1} \left|\sum\limits_{n = N}^\infty  {\frac{1}{{z^n  + 1}}} \right| \ge 
\sum\limits_{n = N}^\infty  {\frac{1}{{\left( {1 + \frac{1}{N}} \right)^n  + 1}}} 
 \ge \frac{1}{{\left( {1 + \frac{1}{N}} \right)^N  + 1}} \ge \frac{1}{{e + 1}}.
$$
