In summary, a Lipschitz function is a type of mathematical function with a finite slope, making it more precise and well-behaved. A series converging to a Lipschitz function implies that the series is approaching a function that is well-behaved and has finite slopes, making it easier to analyze and work with mathematically. The Lipschitz condition is a criterion used to determine if a function is Lipschitz, and a series converging to a Lipschitz function satisfies this condition. It is possible for a series to converge to a Lipschitz function even if the function itself is not Lipschitz, as long as it satisfies the Lipschitz condition.
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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}##.
Set [tex]
f_n(x) = \sum_{k=1}^n \frac{(-1)^{k-1}}{|x| + k}.[/tex] This converges at each [itex]x \in \mathbb{R}[/itex] as [itex]n \to \infty[/itex] by the alternating series test. Let [itex]f(x) = \lim_{n \to \infty} f_n(x)[/itex].
We have [tex]
|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|.[/tex] Now [tex]\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}[/tex] I think the result follows on letting [itex]n \to \infty[/itex].