The series is used to approximate a constant

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

Problem

The series

\displaystyle \sum_{n=1}^{\infty} (-1)^{n+1}\,\frac{3}{n^{2}}

is used to approximate a constant

L

. A student computes the fifth partial sum

S_{5}=\sum_{n=1}^{5} (-1)^{n+1}\,\dfrac{3}{n^{2}}

. (a) Verify that the series satisfies all hypotheses needed for the alternating series test. (b) Let

R_{5}=L-S_{5}

. Explain why

|R_{5}|\le \dfrac{3}{6^{2}}

and interpret the meaning of this inequality in the context of approximating

L

. (c) Without calculating any further numerical value, write an inequality that traps

L

between two numbers using

S_{5}

and the bound obtained in part (b).