Let be the Maclaurin series of an analytic function on an open interval containing .

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

Problem

Let

p(x)=\sum_{k=0}^{\infty}a_kx^k

be the Maclaurin series of an analytic function on an open interval containing

0

. (a) Explain, in complete sentences, why

p^{(k)}(0)=k!\,a_k

for every non-negative integer

k

. Your explanation should reference the definition of the

k

th derivative at

0

and properties of power series. (b) Suppose

a_k=\dfrac{(-1)^k}{k+1}

for all

k\ge0

. Determine

p^{(4)}(0)

and comment on how your result illustrates the relationship discussed in part (a).