Let , and suppose is expanded in a Maclaurin series that converges for . Which of the…

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

p(x)=\dfrac{e^{2x}}{1-x}

, and suppose

p

is expanded in a Maclaurin series that converges for

|x|<1

. Which of the following gives a closed-form expression for

p^{(n)}(0)

for every non-negative integer

n

? (a)

\displaystyle\sum_{k=0}^{n}\frac{2^{k}}{k!}

(b)

n!\,2^{n}

(c)

n!\displaystyle\sum_{k=0}^{n}\frac{2^{k}}{k!}

(d)

\displaystyle\sum_{k=0}^{n}\frac{2^{k}}{(n-k)!}