Show, that $$g(x):=\sum_{k=0}^{\infty} \frac{1}{k ! \cdot\left(1+4^{k} x^{2}\right)}$$ is infinitely often differentiable on $\mathbb{R}$. Furthermore prove that the Taylor Series of $g$ evaluated at zero diverges for all $x\neq 0$.
Expressing the Sum with the function $f:(-1,1) \rightarrow \mathbb{R}, x \mapsto \frac{1}{1+x^{2}}$ might be helpful. Also note $f(x)=\sum_{k=0}^{\infty}(-1)^{k} x^{2 k}$ by the geometric series.