Calculating the Number of Thermal Photons in a Cavity at Equilibrium

[Ignea_unda]

Homework Statement

Show that the number of photons $$\sum_n <s_n>$$ in equilibrium at temperature $$\tau$$ in a cavity of volume V is
$$N = 2.404\pi^{-2}V(\tau / \hbar c)^3$$

Homework Equations

$$<s> = \sum_s s P(s) = Z^{-1} \sum s~ exp[-s \hbar \omega/ \tau] ~~~ (1)$$

$$<s> = 1/exp[\hbar \omega / \tau] - 1] ~~~ (2)$$

$$\omega_n = n \pi c/ L ~~~ (3)$$

The Attempt at a Solution

I have tried to convert omega and take the sum using (3), but haven't had any luck with that. Also I have tried taking the double summation over <s> but have run into a stumbling block with Z being of a similar form of the exact form (2) of <s>.

 

[BigJDubs]

I'm sure I am missing something very simple, but I haven't been able to figure it out. Any help would be appreciated. Thank you.