Taylor Series of Analytic Function has infinite Radius of Convergence

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Theorem

Let $F$ be a complex function.

Let $F$ be analytic everywhere.


Let $f = F {\restriction_{\R}}$ be a real function.

Let $x_0$ be a point in $\R$.


Then the Taylor series of $f$ about $x_0$ converges to $f$ at every point in $\R$.


Proof

The result follows by Convergence of Taylor Series of Function Analytic on Disk for the case $R = \infty$.

$\blacksquare$


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