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Is there any example of a real valued function $f$ defined on an open interval $I$(possibly the real line) that satisfies the following property?

There is an $a\in I$ such that the Taylor series expansion of $f$ relative to $a$ has zero radius of convergence. (The function doesn't have to be equal to its Taylor series in some interval, just thinking about the convergence radius of the Taylor series itself.)

Sphere
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1 Answers1

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By Borel's theorem, every sequence of real numbers may be the coefficients of the Taylor series of some function. If the coefficients are made to be some sufficiently fast-increasing function, the Taylor series diverges at all non-zero values.

Parcly Taxel
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  • ...and $f(0)=0$. – Tito Eliatron Dec 01 '20 at 09:18
  • But isn't $f^{n}(0)=0$ for all $n$? Then the Taylor series is zero function. – Sphere Dec 01 '20 at 09:24
  • @Sphere Yes. Which means that the Taylor series is zero too, but since the function becomes positive on the right of $0$, it has zero radius of convergence. – Parcly Taxel Dec 01 '20 at 09:25
  • Actually, as I wrote in the text, I'm thinking only about the convergence of series itself(whether it equals the original function is not of interest now). In this case, the series itself converges in $\mathbb {R}$ because it is the zero function. – Sphere Dec 01 '20 at 09:28
  • @Sphere I have found an answer - see the duplicate question. – Parcly Taxel Dec 01 '20 at 09:44
  • The Borel theorem you linked seems to be the answer I'm looking for. It says that any sequence of reals can be the taylor coefficients at $x=0$ of a $C-$infinity function, so it is possible that a real valued function $f$ defined on the whole real line has a taylor expansion at $x = 0$ that diverges(this means that the series itself diverges) except $x=0$. Am I right? – Sphere Dec 01 '20 at 09:48
  • @Sphere Yes, yes. – Parcly Taxel Dec 01 '20 at 09:49
  • I guess the following function $$F(x) = \sum_{n=0}^{\infty} \frac {\exp(2^n i x)} {n!} $$ mentioned at https://mathoverflow.net/a/624 serves the purpose. (Never check the detail.) – QA Ngô May 05 '22 at 03:43