Radius of Convergence

The disk where a power series converges, and its boundary.

The idea

The radius of convergence of a power series $\sum_{n=0}^{\infty} a_{n} x^{n}$ describes exactly where the series converges.

Theorem (Radius of convergence).

For every power series $\sum_{n=0}^{\infty} a_{n} x^{n}$ there is a number $R$ with $0 \le R \le \infty$ such that the series converges whenever $|x| < R$ and diverges whenever $|x| > R$.

This $R$ is the radius of convergence: $R = \infty$ means the series converges for every $x$, and $R = 0$ means it converges only at $x = 0$. The ratio test computes $R$. Applied to $\sum a_{n} x^{n}$, it compares consecutive terms; the factor $|x|$ comes out of the comparison, and what remains is a condition on the coefficients alone.

Proposition (Ratio formula for the radius).

If the limit exists, then $\frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_{n}} \right|,$ where a limit of $0$ gives $R = \infty$ and a limit of $\infty$ gives $R = 0$. When the limit does not exist, $\tfrac{1}{R} = \limsup_{n \to \infty} |a_{n}|^{1/n}$ applies in its place.

The radius says nothing about the boundary $|x| = R$, where the ratio test yields exactly $1$ and gives no verdict. We substitute each endpoint into the series and test it as an ordinary series, and the two endpoints can behave differently.

Ways to work on it

• Walkthrough. The disk of convergence and finding R with the ratio test.
• Practice. Find the radius of convergence of a given power series.
• Hardest. Compute the radius, then resolve both endpoints separately.

Not sure where to start? Take the ten-question placement test.