Xyius
This might be more of a mathematical question than a physical one. But I am taking a Quantum Mechanics course and the book starts out by introducing the equation for the energy density of radiation from a black body. They then integrate this expression over infinity to find the total energy per unit volume.

http://img256.imageshack.us/img256/5189/blackbody.jpg [Broken]

My question is, how did they do the integral? It looks like they turned $\frac{1}{e^{x}-1}$ into its geometric series representation. That part I understand. But what do they do in the step after that? Where does the geometric series go? And where does the $\frac{1}{(n+1)^4}$ come from? And for that matter, the last line in the derivation?

I know its not an incredibly crucial question in understanding the Physics, but it bugs me a lot when I cannot follow the mathematics.

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Well when you pull the sum out to the front of the integral, you have $$x^3e^{-(1+n)x}$$. Then when you substitute y = (n+1)x, you have to use x^3 dx= y^3dy / (n+1)^4. The integral can then be evaluated, presumably by parts, to get 6. Evaluating the sum is a bit tricky. If I was working through the derivation, I'd just be satisfied with looking up the answer. This page gives a few clever ways of doing it: http://math.stackexchange.com/questions/28329/nice-proofs-of-zeta4-pi4-90