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Cauchy–Hadamard theorem

A formula for the radius of convergence of a power series using a limsup of nth roots of coefficients.

Cauchy–Hadamard theorem: For a ∑n=0∞an(x−x0)n\sum_{n=0}^\infty a_n (x-x_0)^n (real or complex), define

L=lim sup⁡n→∞∣an∣1/n.L=\limsup_{n\to\infty} |a_n|^{1/n}.

Set the radius of convergence

R=1L,R=\frac{1}{L},

with the conventions 1/0=∞1/0=\infty and 1/∞=01/\infty=0. Then:

  • The series converges absolutely for ∣x−x0∣<R|x-x_0|<R.
  • The series diverges for ∣x−x0∣>R|x-x_0|>R.
Remarks

This result is an application of the to the terms an(x−x0)na_n(x-x_0)^n and is the standard way to compute the radius in practice.