POTW A Series Converging to a Lipschitz Function

[Euge]

Prove that the series $$\sum_{k = 1}^\infty \frac{(-1)^{k-1}}{|x| + k}$$ converges for all $x\in \mathbb{R}$ to a Lipschitz function on $\mathbb{R}$.

 

[pasmith]

Spoiler

Set <br /> f_n(x) = \sum_{k=1}^n \frac{(-1)^{k-1}}{|x| + k}. This converges at each x \in \mathbb{R} as n \to \infty by the alternating series test. Let f(x) = \lim_{n \to \infty} f_n(x).

We have <br /> |f_n(x) - f_n(y)| \leq \sum_{k=1}^n \left|\frac{|y| - |x|}{(|x| + k)(|y| + k)}\right| \leq \sum_{k=1}^n \left|\frac{|y| - |x|}{k^2}\right| &lt; \frac{\pi^2}{6}\left||y| - |x|\right| \leq \frac{\pi^2}{6}|x - y|. Now \begin{split}<br /> |f(x) - f(y)| &amp;\leq |f(x) - f_n(x)| + |f_n(x) - f_n(y)| + |f_n(y) - f(y)| \\<br /> &amp;&lt; \frac{\pi^2}{6}|x - y| + |f(x) - f_n(x)| +|f_n(y) - f(y)|.<br /> \end{split} I think the result follows on letting n \to \infty.