Consider the formal power series in one complex variable $z$ of the form

$$f(z) = \sum_{n = 0}^{\infty} c_{n} (z-a)^{n}$$

where $a,c_n\in\mathbb{C}.$

Then the radius of convergence of $f$ at the point $a$ is given by

$$\frac{1}{R} = \limsup_{n \to \infty} \big( | c_{n} |^{1/n} \big) $$

And confusing for me is "sup". I know what is supremum, but I always ignore it. This is the moment, when I would like understand, so I am asking you for helping. Firstly, why is it necessary?

  • $\begingroup$ Google "lim sup". It is the supremum of the set of partial limits of a sequence. $\endgroup$ – Timbuc Jan 23 '15 at 9:15
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    $\begingroup$ Consider the case $c_n = 1$ or $0$ depends on whether $n$ is even or odd, the limit $\lim_{n\to\infty} |c_n|^{1/n}$ doesn't exist at all. In general, what stop a power series at a particular $z$ from converging are those $c_n$ that glows the fastest at $n \to \infty$. The $\limsup$ allow you to select the contribution from the fastest growing ones. $\endgroup$ – achille hui Jan 23 '15 at 9:20

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