Mathematics

Problem

Define the function

h

h(x)=\frac{x^{2}}{1-3x}

for values of

x

near

0

. a. Obtain a Maclaurin series for

h(x)

and list all terms through and including the power of

x^{4}

. Present your answer in expanded form. b. Use the series from part (a) to evaluate

\int_{0}^{0.1} h(x)\,dx.

Employ enough terms to guarantee that the absolute error of your approximation is less than

10^{-6}

, and explain how your error estimate is obtained.