Show that the series $\sum_{n=1}^\infty \frac{1}{n^2} \cos( \frac{x}{n^2})$ converges uniformly

by Mathias   Last Updated May 22, 2020 23:20 PM - source

For $x \in \mathbb{R}$ consider the series $$ S = \sum_{n=1}^\infty \frac{1}{n^2} \cos( \frac{x}{n^2}) $$ Then I have to show that $S$ converges uniformly. I think I have to use Weiterstrass M-test but I am not sure whether or not that $$ \left| cos(\frac{x}{n^2}) \right| \leq 1 $$ is true for all $x \in \mathbb{R}$ and for all $n \in \mathbb{R}$. I know that $$ \left| sin(\frac{x}{n^2}) \right| \leq \frac{|x|}{n^2} $$ but I don't think it is the same for cosine? Am I able to say that $$ \left| \frac{1}{n^2} \cos( \frac{x}{n^2}) \right| \leq \frac{1}{n^2} $$ Thus as $\sum_{n=1}^\infty \frac{1}{n^2}$ converges $S$ converges uniformly by Weiterstrass' M-test.

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Updated July 05, 2020 05:20 AM