AP

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

H(x)=\dfrac{x^{2}}{(1-x)^{3}}

for

x\neq 1

. (a) Express

H(x)

as a power series centered at

x=0

by writing a general term for the series. (b) Provide a closed-form expression for the coefficient of

x^{n}

in your series for all

n\ge 0

. (c) Evaluate the infinite series

\sum_{n=2}^{\infty} n(n-1)\left(\frac{1}{3}\right)^{n}

by appropriately substituting into your series from part (a).