The Reimann zeta$-$function is defined by

$($x$)=$

$$

n$=1$

$1$

nx

and is used in number theory to study the distribution of primes.

$($a$)$ For values of x does converge? $($i$.$e What is the domain of $?)$ Explain your reasoning.

$($b$)$ Leonhard Euler was able to calculate the exact sum of the p$-$series with p $=2$ and p $=4$

$(2)=$

$$

n$=1$

$1$

n$2=$ $2$

$6$ $(4)=$

$$

n$=1$

$1$

n$4=$ $4$

$90$

Use Euler$$s results to find the sum of the following series:

I.

$$

n$=3$

$1$

$($n $2)2$

II$.$

$$

n$=1$

$(3$

n $)$

$4$

III.

$$

n$=2$

$1$

n$2$

IV$.$

$$

n$=1$

$1$

$($n $+3)4$