AP

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

Problem

Let

F(x)=e^{\sin(2x)}

. (a) Obtain the Maclaurin series for

F

and list all terms up to and including the term containing

x^{5}

. (b) State explicitly the coefficient of

x^{5}

in the series. (c) Determine the radius of convergence of the Maclaurin series for

F

and justify your answer. (d) Discuss whether the interval of convergence of the series for

F

is identical to that of the Maclaurin series for

e^{x}

. Support your conclusion with reasoning based on properties of series and composition of functions.