AP Calculus BC glossary

Binomial series

The binomial series is the Maclaurin expansion of one plus x raised to a real power k. When k is a non-negative integer it terminates and reproduces the binomial theorem; otherwise it is an infinite series converging for x between negative one and one.

(1+x)k=1+kx+k(k−1)2!x2+k(k−1)(k−2)3!x3+⋯(1+x)^{k} = 1 + kx + \frac{k(k-1)}{2!}x^{2} + \frac{k(k-1)(k-2)}{3!}x^{3} + \cdots

Taking k=12k = \frac{1}{2} gives the series for 1+x\sqrt{1+x}, and k=−1k = -1 gives the geometric series, which is why the geometric series is a special case rather than a separate fact.

The mistake

Expecting it to terminate for a negative or fractional exponent. The factors never hit zero in that case, so the series genuinely runs forever.

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