AP Calculus BC glossary

Factorial

Also called: n factorial

The factorial n! is the product of the positive integers from 1 up to n, and 0! is defined to be 1. The fact that settles most series questions is that (n+1)! equals (n+1) times n!, which is why factorials cancel down to a single factor in the ratio test.

n!=n(n−1)⋯2⋅1,0!=1n! = n(n-1)\cdots 2 \cdot 1, \qquad 0! = 1

Peeling one factor off the front is the whole technique: (n+1)!=(n+1)⋅n!(n+1)! = (n+1)\cdot n!. Divide consecutive terms of a series and the two factorials collapse into one factor rather than two long products.

∣xn+1/(n+1)!xn/n!∣=∣x∣n+1→0\left|\frac{x^{n+1}/(n+1)!}{x^{n}/n!}\right| = \frac{|x|}{n+1} \to 0

That limit is 00 for every xx, so ∑xn/n!\sum x^n/n! converges on the whole real line, which is why the series for exe^x has infinite radius of convergence. Factorial growth beats exponential growth: n!n! eventually passes cnc^n for any fixed cc. The convention 0!=10! = 1 lets the constant term of a Maclaurin series be written in the same form as every other term. One more piece of notation: (2n)!(2n)! is the product of every integer up to 2n2n, which is not 2⋅n!2 \cdot n!, and at n=3n = 3 that is 720720 against 1212.

The mistake

Advancing (2n)!(2n)! by one index step multiplies by two factors, not one: (2n+2)!=(2n+2)(2n+1)(2n)!(2n+2)! = (2n+2)(2n+1)(2n)!. Writing (2n+2)!=(2n+2)(2n)!(2n+2)! = (2n+2)(2n)! is the usual reason a ratio test on terms built from (2n)!(2n)! comes out wrong.

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