Consider the series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^2}$

$($a$)$ Find the tenth partial sum, $s_{10}.($Round your answer to six decimal places.$)$

$s_{10}=$

Estimate the error in using $s_{10}$ as an approximation to the sum of the series.

$R_{10}{\displaystyle \underset{10}{\overset{}{}}}\frac{1}{x^2}dx=$

$$

$($b$)$ Use the following inequalities to obtain an improved estimate of the sum. $($Round your answers to six decimal places.$)$

$s_n{\displaystyle \underset{n1}{\overset{}{}}}f(x)dxs{s}_n{\displaystyle \underset{n}{\overset{}{}}}f(x)dx$

$5$s $5$

$($c$)$ Find a value of $n$ so that $s_n$ is within $0\mathrm{.}001$ of the sum.

$n>5$

$n>100$

$n>1000$

$n>-1000$

$n>0$