Cauchy-Hadamard theorem

[edit] Statement of the theorem

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 is given by

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

where the limsup is defined as

$\limsup_{n\to\infty} u_n:=\lim_{n\to\infty}\sup\{u_k:k\leq n\}\leq 0$

where the sup is the least upper bound of a set.

Note:
$\frac{1}{0} := \infty$
for this Theorem (and vice versa).

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