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