AP Calculus BC glossary

Maclaurin series

A Maclaurin series is a Taylor series centred at zero. It is not a different object, just the most common special case, and the standard ones for the exponential, sine, and cosine functions are worth knowing by heart.

f(x)=∑n=0∞f(n)(0)n!xnf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n

The four to memorize are ex=∑xnn!e^x = \sum \frac{x^n}{n!}, sin⁡x=∑(−1)nx2n+1(2n+1)!\sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}, cos⁡x=∑(−1)nx2n(2n)!\cos x = \sum \frac{(-1)^n x^{2n}}{(2n)!}, and 11−x=∑xn\frac{1}{1-x} = \sum x^n for ∣x∣<1|x| < 1.

Almost every series question builds from those four by substitution, multiplication by a power, differentiation, or integration, which is far faster than computing derivatives one at a time.

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