Mathematics

Problem

The convergent series

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

is to be approximated numerically. a. For an integer

N\ge 1

, evaluate the improper integral

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

in simplest form. b. 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. c. 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.