Let the functions and be represented by Maclaurin series f(x)= _ n=0 ^ a_ n x^ n

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

Problem

Let the functions

f

and

g

be represented by Maclaurin series

f(x)=\sum_{n=0}^{\infty}a_{n}x^{n},\qquad g(x)=\sum_{n=0}^{\infty}b_{n}x^{n},

with radii of convergence

R_{f}

and

R_{g}

, respectively. Define

h(x)=f(x)g(x)

. (a) Derive an explicit formula for the coefficient

c_{n}

in the Maclaurin series

\displaystyle h(x)=\sum_{n=0}^{\infty}c_{n}x^{n}

in terms of

a_{k}

and

b_{n-k}

. Justify each step in your derivation. (b) Prove that the series for

h(x)

converges absolutely for every

x

satisfying

|x|<\min(R_{f},R_{g})

. Clearly state any theorems you invoke. (c) Explain why the radius of convergence of the Maclaurin series for

h

must be at least

\min(R_{f},R_{g})

, and describe a situation in which the radius could be strictly larger.