Mathematics

Problem

Consider the function

f(x)=x^{2}\cos\!(x^{3})

. a. Substitute

x^{3}

into the Maclaurin series for

\cos x

and write the first four non-zero terms of the resulting series for

\cos\!(x^{3})

. Express each term in powers of

x

. b. Multiply your result from part (a) by

x^{2}

to obtain the first four non-zero terms of the Maclaurin series for

f(x)

. c. Without further calculation, describe the complete set of exponents that appear in the Maclaurin series for

f(x)

. Then write a closed-form expression for the general coefficient

a_{3k+2}

x^{3k+2}

, where

k\ge 0

is an integer. Briefly justify your formula based on the structure of the series you obtained.