What is the Function for the Value of a Convergent Series Sum?

[mathman]

TL;DR

sum $\frac{1}{n^c}$ where $c\gt 1$

$\sum_n \frac{1}{n^c}$ converges for $c\gt 1$. Is there an expression for the value of the sum as a function of $c$?

 

[mfb]

For some values there are analytic expressions. It's the Riemann zeta function.

 

[member 587159]

No, there is not. A closed expression for $\sum_{n=1}^\infty \frac{1}{n^3}$ is not known. This is Apéry's constant. See https://en.wikipedia.org/wiki/Apéry's_constant for more information.

 

[mathman]

For some values there are analytic expressions. It's the Riemann zeta function.

I should have known! It is the zeta function for all $c\gt 1$.