Homework$-8$: Problem $13$

$(5$ points$)$

The series

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{(4n+2{)}^2}$

is convergent.

$($A$).$ According to the Remainder Estimate for the Integral Test, the error in the approximation $s$~~$s_n($where $s$ is the value of the infinite sum and $s_n$ is the $n-$th partial sum$)$ is

$|s-s_n|$

$($B$).$ Find the smallest value of $n$ such that this upper bound is less than $0\mathrm{.}0008.$

$n=$