Convergent power series/R/Taylor series/Coincidence/Fact

Let ${}\sum _{n=0}^{\infty }c_{n}x^{n}$ denote a power series which converges on the interval ${}]-r,r[$, and let

$f\colon ]-r,r[\longrightarrow \mathbb {R}$

denote the function defined via fact.

Then ${}f$ is infinitely often

differentiable, and the Taylor series

of ${}f$ in ${}0$ coincides with the given power series.

Proof, Write another proof