The convergent series is to be approximated numerically.

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

Problem

The convergent series

\displaystyle S=\sum_{n=1}^{\infty}\dfrac{1}{n^{2}}

is to be approximated numerically. (1) For an integer

N\ge 1

, evaluate the improper integral

\displaystyle\int_{N}^{\infty}\dfrac{1}{x^{2}}\,dx

in simplest form. (2) Let

R_N=S-\displaystyle\sum_{n=1}^{N}\dfrac{1}{n^{2}}

. Using the integral you found in part (a), determine the smallest integer

N

for which

R_N<0.05

. Show your reasoning clearly. (3) Compute

\displaystyle\sum_{n=1}^{N}\dfrac{1}{n^{2}}

for the

N

obtained in part (b). State an interval of length

0.05

that is guaranteed to contain

S

. Explain why your interval is valid.