I know that a power series with radius of convergence $R>0$ converges uniformly on $[-R_1,R_1]$ to a continuous function (where $0<R_1<R$). Would that imply that if $R=+\infty$, the power series would converge uniformly everwhere?
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asked Dec 13, 2013 at 21:13
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3$\begingroup$ No, only that it converges uniformly on every bounded interval. Consider the power series for $\sin$, for example. Every partial sum is unbounded, but $\sin$ is bounded on $\mathbb{R}$. $\endgroup$– Daniel FischerCommented Dec 13, 2013 at 21:15
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No, it does not: The Taylor series for $e^x$ converges pointwise everywhere, but does not do so uniformly. We have
$$e^x = \sum\limits_{k = 0}^{\infty} \frac{x^k}{k!}$$
One way to see that the convergence is not uniform is to let $x \to -\infty$; then $e^x$ is bounded, while every Taylor polynomial tends to one of $\pm \infty$ as $x$ decreases.