Radius of Convergence — Mathematics Knowledge System

Overview

Every power series sum_{n=0}^inf a_n*(x-c)^n has a radius of convergence R >= 0 (possibly R = inf) such that the series converges absolutely for |x-c| < R and diverges for |x-c| > R. At |x-c| = R (the boundary), convergence must be checked separately. The radius is given by the Cauchy-Hadamard formula: 1/R = limsup_{n->inf} |a_n|^{1/n}, or by the ratio test: R = lim_{n->inf} |a_n/a_{n+1}| when this limit exists.

Intuition

A power series centered at c converges in a 'disk' (or interval in 1D) of radius R around c and diverges outside. You can think of R as how far from the center the series can 'see' before breaking down. The boundary circle |x-c| = R is where it gets complicated: the series might converge, diverge, or even conditionally converge depending on the specific point. The ratio test gives R as the limit of |a_n|/|a_{n+1}|: if successive coefficients decrease fast, R is large.

Formal Definition

Definition

For sum a_n*(x-c)^n, define R by 1/R = limsup_{n->inf} |a_n|^{1/n} (Cauchy-Hadamard). Conventions: if limsup = 0 then R = inf (everywhere convergent, like e^x); if limsup = inf then R = 0 (only at x=c). Ratio test form: if L = lim|a_{n+1}/a_n| exists, then R = 1/L. The interval of convergence is (c-R, c+R) plus any endpoints where the series converges (test separately using e.g. alternating series test or p-series).

\frac{1}{R} = \limsup_{n\to\infty}|a_n|^{1/n}

Cauchy-Hadamard formula

R = \lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right| \text{ (when limit exists)}

Ratio test form

\sum a_n(x-c)^n \text{ converges absolutely for } |x-c|<R

Convergence region

Notation

Notation	Meaning
$R$	Radius of convergence
$\limsup$	Limit superior of a sequence
$(c-R,\, c+R)$	Interval of convergence (open part)

Theorems

Theorem 1: Cauchy-Hadamard Theorem

The radius of convergence of sum a_n*(x-c)^n is R = 1/limsup|a_n|^{1/n}, with 1/0 = inf and 1/inf = 0. The series converges absolutely for |x-c| < R, diverges for |x-c| > R, and may converge or diverge at |x-c| = R.

Theorem 2: Uniform Convergence on Compact Subsets

If a power series sum a_n*(x-c)^n has radius of convergence R > 0, then it converges uniformly on any closed interval [c-r, c+r] with r < R. In particular, the sum function is continuous and infinitely differentiable on (c-R, c+R), and can be differentiated/integrated term by term.

Worked Examples

1
a_n = 1/n. Ratio test: |a_{n+1}/a_n| = (1/(n+1))/(1/n) = n/(n+1) -> 1 as n->inf.
$\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = \lim_{n\to\infty}\frac{n}{n+1} = 1$
2
So 1/R = 1, giving R = 1. At x=1: sum 1/n diverges (harmonic). At x=-1: sum (-1)^n/n converges (alternating). IOC: [-1, 1).
$R = 1,\quad \text{IOC} = [-1,1)$

✓ Answer

R = 1. Interval of convergence [-1, 1) (converges at x=-1 by alternating series test, diverges at x=1).

Practice Problems

Mediumapplication

Find the radius and interval of convergence of sum_{n=0}^inf n!*x^n.

Common Mistakes

Common Mistake

Concluding convergence at boundary from R alone.

Knowing R tells you absolute convergence for |x-c| < R and divergence for |x-c| > R, but says nothing about the boundary |x-c| = R. Must check each endpoint separately. The series sum x^n/n converges at x=-1 (alternating series) but diverges at x=1 (harmonic series). Always test both endpoints when R is finite.

Quiz

The power series for e^x = sum x^n/n! has radius of convergence:

R = 1R = eR = infinityR = 0

Historical Background

The theory of power series and their convergence was developed rigorously in the 19th century. Cauchy gave convergence criteria for series (1821). Hadamard (1892) proved the formula 1/R = limsup|a_n|^{1/n}. Abel's theorem (1826) established that if sum a_n converges, the power series sum a_n*x^n converges uniformly on [0,1] to the same sum. The concept of radius of convergence is central to complex analysis (analytic functions have power series with a disk of convergence).

1821
Cauchy develops rigorous convergence theory for series
Augustin-Louis Cauchy
1826
Abel proves his theorem on uniform convergence at boundary
Niels Henrik Abel
1892
Hadamard proves the radius formula via limsup
Jacques Hadamard

Summary

Radius R: Cauchy-Hadamard formula 1/R = limsup|a_n|^{1/n}; ratio form R = lim|a_n/a_{n+1}|.
Converges absolutely for |x-c| < R, diverges for |x-c| > R; test endpoints separately.
Within radius: series is infinitely differentiable; differentiate/integrate term by term.
R = 0 (only at center), R = inf (everywhere): e.g. n!*x^n vs e^x = sum x^n/n!.

References

BookRudin, W. Principles of Mathematical Analysis. 3rd ed. McGraw-Hill, 1976. Ch. 3.
WebsiteWikipedia -- Radius of convergence