Cauchy-Hadamard Theorem/Real Case

Theorem

Let $\xi \in \R$ be a real number.

Let $\ds \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about $\xi$.

Then the radius of convergence $R$ of $S \paren x$ is given by:

$\ds \frac 1 R = \limsup_{n \mathop \to \infty} \size {a_n}^{1/n}$

If:

$\ds \frac 1 R = \limsup_{n \mathop \to \infty} \size {a_n}^{1/n} = 0$

From the $n$th root test, $S \paren x$ is convergent if $\ds \limsup_{n \mathop \to \infty} \size {a_n \paren {x - \xi}^n}^{1/n} < 1$.

Thus:

$\ds \size {a_n \paren {x - \xi}^n}^{1/n}$

$<$

$\ds 1$

$\ds \leadstoandfrom \ \ $

$\ds \size {a_n}^{1/n} \size {x - \xi}$

$<$

$\ds 1$

$\ds \leadstoandfrom \ \ $

$\ds \size {a_n}^{1/n}$

$<$

$\ds \frac 1 {\size {x - \xi} }$

The result follows from the definition of radius of convergence.

$\blacksquare$