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Just a random thought. Why, intuitively, is it that a Taylor series in one variable, WLOG centered at $0$, either converges everywhere or in a finite disk/interval centered at $0$ (with some subtleties on the boundary) - as opposed to some arbitrary neighborhood of $0$?

I realize this can be mathematically shown by means of the root test or something but, I was wondering whether there was a more intuitive explanation.

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    Are you talking about power series in one variable or in several variables? Real or complex? [In several variables, the domain of convergence can be way more complicated than a ball, in one variable, the use of the word "ball" is unusual, more common would be "disk" in the case of one complex variable and interval in the real case.] – 2017-02-02
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    Yes, let's go with disk. I'll make the edit – 2017-02-02
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    The set of $z$ such that $a_n\cdot z^n \to 0$ is a disk (either open or closed, possibly the entire plane, possibly just $\{0\}$, we regard the former as a disk with infinite radius and the latter as a closed disk with radius $0$ to have a uniform terminology). That's very easy to see. And for a power series, the terms converging to $0$ almost implies convergence [$a_nw^n$ being bounded implies that the series converges for all $z$ with $\lvert z\rvert < \lvert w\rvert$]. – 2017-02-02

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