uniform convergence of a series

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

  1. Pete

    Pete

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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
    #1
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