Mathematics

Problem

A function

f

has Maclaurin series

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

with first four nonzero coefficients given by

a_0=0,\;a_1=1,\;a_2=-\dfrac12,\;a_3=\dfrac13

. Define

g(x)=x\,[f(x)]^2

. (a) Write the Maclaurin series for

g(x)

through the term in

x^4

. Clearly list the coefficients you obtain. (b) Determine the exact value of

g''(0)

. Show how the coefficient you found in part (a) leads to your answer.