Let $\sum_{n=0}^\infty a_n(x-c)^n$ be a power series with $\infty$ radius of convergence. In this case, is there any conclusion that will $\sum_{n=0}^\infty a_n(x-c)^n$ converges uniformly on $\Bbb R$? (I can't find such theorem in my analysis books.) For example, is $1+x+\frac{1}{2!}x^2+\cdots$ converges uniformly on $\Bbb R$ (not just on a closed bounded interval)?
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4$\begingroup$ In general the series is not uniformly convergent in $\mathbb{R}$. In your example, the sum is $f(x) = e^x$, and $\sup |e^x - s_n(x)| = +\infty$ for every $n$ (being $s_n$ the partial sum). $\endgroup$– RigelDec 10, 2017 at 12:49
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No, unless it is a polynomial, the summands don't converge to $0$ uniformly on $\Bbb R$ (if $a_n$ is nonzero, then $a_nx^n$ is unbounded), so the series doesn't converge uniformly on $\Bbb R$.
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$\begingroup$ Why does $a_nx^n$ unbounded imply not being uniformly convergent? $\endgroup$– EricDec 10, 2017 at 13:31
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1$\begingroup$ If $S_n$ converge uniformly, $S_{n+1}$ also, so the difference converges uniformly to $0$ by triangle inequality. $\endgroup$– user371663Dec 10, 2017 at 15:52
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