The Maclaurin series for is given by for

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

Problem

The Maclaurin series for

\arctan x

is given by

\arctan x = \displaystyle\sum_{m=0}^{\infty} \dfrac{(-1)^{m}x^{2m+1}}{2m+1}

for

|x|\le 1

. This series evaluated at

x=1

yields

\dfrac{\pi}{4}

. (a) Write the expression for the tenth partial sum

T_{10}

obtained by keeping the first ten non-zero terms of the series at

x=1

. Give your answer as an exact rational combination of powers of

1

. (b) Using only information about the size of the next unused term, explain why the magnitude of the error in using

T_{10}

to approximate

\dfrac{\pi}{4}

is less than

\dfrac{1}{21}

. (c) Convert the error bound from part (b) to a decimal and determine the number of decimal places in

T_{10}

that are guaranteed to agree with the decimal expansion of

\dfrac{\pi}{4}

. Justify your conclusion.