The Maclaurin series for evaluated at gives Determine the smallest integer so that…

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

The Maclaurin series for

\arctan x

evaluated at

x=1

gives

\displaystyle \frac{\pi}{4}= \sum_{n=0}^{\infty} (-1)^{n}\dfrac{1}{2n+1}.

Determine the smallest integer

N

so that the partial sum

S_N=\displaystyle\sum_{n=0}^{N} (-1)^{n}\dfrac{1}{2n+1}

approximates

\dfrac{\pi}{4}

with an absolute error less than

1\times10^{-4}