# 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

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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