Suppose $f$ is $C^\infty$ on an open set $U$, and $x \in U$. Does $Tf$, the Taylor series centered at $x$, converge in a neighborhood of $x$ to some function(not necessarily the original function $f$)? To be more precise, if $B$ is an open ball centered at $x$ contained in $U$, do we have $Tf$ converges in $B$?
I’m not asking about Taylor series not equal to the original function. It’s a totally different question.
Best Answer
Yes, such functions exist. Take a power series centered at $0$ such that its radius of convergence is $0$, such as, say,$$\sum_{n=0}^\infty n^nx^n.\tag1$$Then, according to Borel's lemma, there is a smooth function $f\colon\Bbb R\longrightarrow\Bbb R$ such that$$(\forall n\in\Bbb N):f^{(n)}(0)=n!n^n.$$So, the Taylor series of $f$ centered at $0$ is the power series $(1)$.