I've been working on this problem regarding uniform convergence of a sum of functions.
Let $g':(-1,1) \to \mathbb{R} $ with $g'$ being bounded at $f(0) = 0$. Show that $S(x) = \sum_{n=1}^{\infty} \frac{1}{n}g(\frac{x}{n+1}) $ converges uniformly on $(-1,1)$.
I know a potential avenue in proving this proposition may be involve the usage of power series and showing that the interval convergence of is $(-1,1)$, thus establishing the proposition.
Another avenue that I've thought is using the Weierstrass M- Test, however I'm not able to construct a sequence that satisfies the condition.
I would prefer hints on approaching this problem, thanks.
\begin{align*} \dfrac{1}{n}\left|g\left(\dfrac{x}{n+1}\right)\right|&\leq\dfrac{1}{n}\int_{0}^{x/(n+1)}|g'(t)|dt\\ &\leq\dfrac{M}{n}\dfrac{|x|}{n+1}\\ &\leq\dfrac{M}{n(n+1)}, \end{align*} where $M$ is the bound for $|g'|$, now invoke Weierstrass M-test.