Cauchy-Hadamard Theorem

analysis

Definition

Cauchy-Hadamard Theorem

The convergence radius $R$ of a power series ${\sum }_{n=0}^{\infty }{a}_n(x-x_0{)}^n$ is determined by the limit superior of the absolute values of the coefficients:

$$\frac{1}{R}=\underset{n\to \infty }{\mathrm{limsup}}\sqrt[n]{|a_n|}$$

where:

• If ${\mathrm{limsup}}_{n\to \infty }\sqrt[n]{|a_n|}=0$, then $R=\infty$.
• If ${\mathrm{limsup}}_{n\to \infty }\sqrt[n]{|a_n|}=\infty$, then $R=0$.

The theorem ensures that the series converges absolutely for all $|x-x_0|<R$ and diverges for all $|x-x_0|>R$. It is a direct consequence of the Root Criterion.