Power Series: Radius and Interval of Convergence

A power series $\sum_{n=0}^\infty c_n (x-a)^n$ is an infinite polynomial centred at $a$. For each fixed $x$, it is a number series whose convergence can be tested. The key result — the Radius of Convergence Theorem — says that the set of $x$-values for which a power series converges is always an interval centred at $a$, with half-length $R$ (the radius of convergence). For $|x-a| < R$ the series converges absolutely; for $|x-a| > R$ it diverges; the boundary requires case-by-case testing. Radius of convergence governs the domain of validity for Taylor approximations of bond prices, option Greeks, and yield-to-maturity expansions — using the expansion outside its radius of convergence gives wrong answers.

Learning objectives. By the end of this note you should be able to: