Consider the power series F(x)= _ k=0 ^ 2^ k ,x^ 3k (3k)!

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

Problem

Consider the power series

F(x)=\sum_{k=0}^{\infty}\frac{2^{k}\,x^{3k}}{(3k)!}.

Assume the series converges for all real

x

. (a) Express the general coefficient

A_n

in the expansion

F(x)=\sum_{n=0}^{\infty}A_nx^{n}

in terms of

n

. Indicate for which values of

n

the coefficient is non–zero. (b) Compute

F^{(9)}(0)

and explain how your answer follows from the coefficient identified in part (a). (c) Find a closed‐form expression for

F(x)

written as an elementary combination of standard functions, and justify your result using series properties.