a$)$ Use the Integral Test to show that

$$

n$=1$

$1/$n$4$ converges.

$($b$)$ Find the $10$th partial sum, s$10,$ of the series

$$

n$=1$

$1/$n$4.$

$($c$)$ According to the Remainder Estimate for the Integral Test, we know that

$$

n$+1$

$1$

x$4$ dx $$ s $$ sn $$

n

$1$

x$4$ dx$,(1)$

where s is the sum of

$$

n$=1$

$1/$n$4$ and sn is the nth partial sum of

$$

n$=1$

$1/$n$4.$ Use inequality

$(1)$ and s$10$ from part $($b$)$ to find an upper and lower bound for s$.$

$($d$)$ Use the Remainder Estimate for the Integral Test to find a value of n so that sn is within

$0\mathrm{.}00001$ of the sum.