Analytic functions are those that are differentiable infinitely.
let $f$ be a function. My claims are as follows :
C1 : Functions that are analytic are exactly those who have taylor series expansions which converges for all x. (Radius of converges infinite)
C2 : If $f$ is an analytical functions (it's differentiable infinitely) I can create it's taylor series expansion around any point $x=x_0$ such that radius of convergence is infinite.
Are my Claims true? false?
Best Answer
Hardly anything you wrote is true.
First of all, an analytic function is not the same thing as a function which has derivatives of all orders. It is stronger than that: it is a function $f$ such that, for each $x_0$ in its domain, the Taylor series of $f$ converges to $f$ in some interval $(x_0-\varepsilon,x_0+\varepsilon)$.
Besides, the function $f\colon\mathbb{R}\longrightarrow\mathbb R$ defined by $f(x)=\frac1{1+x^2}$ is analytic, but the radius of convergence of its Taylor series at $0$ is $1$, not $+\infty$. And, of course, this shows that your second claim is also false.