AP Calculus BC glossary

Power series representation

Also called: Series representation

A power series representation writes a function as an infinite sum of powers of x minus a centre, valid on an interval around that centre. Most are built by substituting into the geometric series, or by differentiating or integrating a series you already know.

11−x=∑n=0∞xn,∣x∣<1\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n, \qquad |x| < 1

Substituting changes the function, and the convergence condition has to be recomputed from the new ratio: sometimes it lands back on the same interval, sometimes not. Replacing xx with −x2-x^2 gives 11+x2=∑n=0∞(−1)nx2n\frac{1}{1+x^2} = \sum_{n=0}^{\infty} (-1)^n x^{2n}, where ∣−x2∣<1|-x^2| < 1 reduces to ∣x∣<1|x| < 1 again, and integrating that term by term produces the series for arctan⁡x\arctan x.

To force a function into geometric shape, get it to a1−r\frac{a}{1-r} with the variable inside rr. Writing 13+x\frac{1}{3+x} as 13⋅11−(−x3)\frac{1}{3} \cdot \frac{1}{1 - \left(-\frac{x}{3}\right)} gives ∑n=0∞(−1)nxn3n+1\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{3^{n+1}} for ∣x∣<3|x| < 3.

The mistake

Keeping the old interval after a substitution. The geometric series needs its ratio under 11 in size, so replacing xx with 2x2x forces ∣2x∣<1|2x| < 1, an interval of ∣x∣<12|x| < \frac{1}{2} rather than ∣x∣<1|x| < 1.

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