AP Calculus BC glossary

Center of a power series

Also called: Center of convergence

The center of a power series is the fixed value the series is built around, the number c in a sum of terms times x minus c to the nth power. The series always converges at its center, and its interval of convergence is symmetric about it, reaching one radius in each direction.

A Taylor series centred at $a$ uses $a$ as its center, and a Maclaurin series is the special case with center $0$ . Picking a center near where you actually want to evaluate makes the polynomial approximations accurate with fewer terms.

Because convergence is symmetric, the interval runs from $c - R$ to $c + R$ , where $R$ is the radius of convergence. The two endpoints must still be checked one at a time, since the series can converge at one, both, or neither.

The mistake

Reading the center's sign wrong. In $(x - c)^n$ the center is $c$ , so $\sum (x + 3)^n$ is centred at $-3$ , not $+3$ . The sign flips because the factor is $x$ minus the center.

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