AP Calculus BC glossary

Power series

A power series is an infinite sum of constant multiples of powers of the variable minus a fixed centre. It converges for the centre always, and generally on an interval around it whose half-width is the radius of convergence.

∑n=0∞cn(x−a)n\sum_{n=0}^{\infty} c_n (x - a)^n

The value of xx decides everything, so a power series is really a family of series. Finding where it converges is the first question asked about one.

Inside the interval of convergence a power series can be differentiated and integrated term by term, and the radius stays the same, though the endpoint behaviour can change.

Appears in: Unit 10: Infinite Sequences and Series (BC)