[Math] Infinitely differentiable function with divergent Taylor series

Question

I'd greatly appreciate it if someone could provide examples of the following:

1) A infinitely differentiable function whose Taylor series does not converge to the function.

2) An infinitely differentiable function whose Taylor series diverges.

My differential equations text says that "most" smooth functions are of one of these types. What can be done if we weaken the infinitely differentiable condition?

(Please do not use $f(x) = e^{-1/x^2}$ as an example!)

Answer 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.