AP

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

Problem

Let

T=\displaystyle \sum_{n=1}^{\infty} (-1)^{n}\,\dfrac{1}{2n-1}

. (a) Compute the sixth partial sum

S_{6}=\sum_{n=1}^{6} (-1)^{n}\,\dfrac{1}{2n-1}

and give its decimal approximation to four decimal places. (b) Using

S_{6}

, determine numbers

A

and

B

such that

A< T< B

. State

A

and

B

explicitly. (c) Find the width

B-A

of the interval you obtained in part (b) and interpret its meaning in terms of the quality of the approximation

S_{6}