# uniform convergence of a series

Discussion in 'Analysis and Topology' started by Pete, Dec 22, 2021.

1. ### Pete

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Dec 22, 2021
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Hi folks,

Let $p>1$. I've to show that $\sum_{n=1}^\infty \frac{\pi}{2} - \arctan(n^p(1+x^2))$ converges uniformly in $\mathbb{R}$.

Is my approach ok?

1.Approach:

Let $\epsilon>0$ be given. Since $\arctan(x)$ converges to $\pi/2$ for $x \rightarrow \infty$ we can find some positive integer $N$ such that $|\frac{\pi}{2} -\arctan(n^p(1+x^2))|<\frac{\epsilon}{m-n+1} for all$m>n>N$. Thus$|\sum_{n=1}^\infty \frac{\pi}{2} - \arctan(n^p(1+x^2))|\leq \sum_{n=1}^\infty |\frac{\pi}{2} - \arctan(n^p(1+x^2))|<\epsilon$for all$ m>n>N\$.

Pete, Dec 22, 2021