A student is investigating the product function

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

Problem

F(x)=e^{x}\cos x

. Both component functions admit well–known Maclaurin series. Answer the following questions about the Maclaurin series of

F

. (a) Explain how a general term of the Maclaurin series for

F

can be obtained by combining terms of the series for

e^{x}

and

\cos x

. State the form of the coefficient of

x^{n}

in terms of a finite sum (you do not need to simplify the sum). (b) Determine the radius of convergence of the Maclaurin series for

F(x)

. Justify your answer clearly. (c) Discuss whether the interval of convergence of the series for

F

is the same as, larger than, or smaller than the intervals of convergence of the series for

e^{x}

and

\cos x

. Provide reasoning based on part (b) and on properties of power series under multiplication.