Infinitely differentiable function with divergent Taylor series?

Solution 1:

Infinitely differentiable functions whose Taylor series diverges except at $0$:

1. $\displaystyle f(x)=\int_0^\infty e^{-t}\cos(t^2 x)\;dt$.

Source: A primer of real functions by R. Boas Jr.

2. $\displaystyle f(x)=\sum_{n=1}^\infty f_n(x)$, where $f_n(x)=\phi_{n,n-1}(x)$, where $$\begin{aligned} &\phi_{n1}(x)=\int_0^x\phi_{n0}(t)\;dt,\\ &\phi_{n2}(x)=\int_0^x\phi_{n1}(t)\;dt,\\ &\quad\vdots\\ &\phi_{n,n-1}(x)=\int_0^x\phi_{n,n-2}(t)\;dt, \end{aligned}$$ where $$\phi_{n0}(x)=\left\{\begin{aligned} ((n-1)!)^2,&\quad \text{if}\quad 0\leq |x|\leq \frac{1}{2^{n}(n!)^2}\\ 0,&\quad \text{if}\quad |x|\geq \frac{1}{2^{n-1}(n!)^2}. \end{aligned}\right.$$

Source: Counterexamples in Analysis by B. Gelbaum and J. Olmsted.