Let $g$ be a smooth function with all n-th derivatives satisfying $\vert g^{n}(x)\vert\leq \frac{C n!}{\rho^n}$ where $\rho$ is a fixed number for all $x\in R$. $f(x)=\sum_n \frac{g^{(n)}(0)}{n!}(x)^n$. Is this series converges to $f(x)$ uniformly on $R$? I am looking for a counter example as I do not think it holds.
2026-03-27 19:10:12.1774638612
Do taylor series smooth functions of power series with finite convergence radius at every point converge to itself on every point of real line
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