Mathematics

Problem

The Maclaurin series for

\arctan x

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

. Let

T

be the sum of the first four nonzero terms of this series evaluated at

x=\dfrac13

. Is

T

guaranteed to approximate

\arctan\!\left(\dfrac13\right)

correctly to three decimal places? a) Yes, because the maximum possible error is exactly

0.0005

. b) No, because the maximum possible error exceeds

0.0005

. c) Yes, because the maximum possible error is less than

0.0005

. d) No, because the series does not alternate in sign for

x=\dfrac13