Deriving the Stefan-Boltzman law and integration tricks

[azoroth134]

Homework Statement

HI people,

I was trying to derive the stefan-Boltzmann law from the planc's formula, I kind of got stuck with an integral

Homework Equations

$$ \int_{0}^{\infty} \frac{x^3}{e^x -1} dx $$

The Attempt at a Solution

I tried simplifying it with

$$ \int_{0}^{\infty} x^3 e^{-x} \sum_{n=0}^{\infty} e^{-nx} dx $$

Now I don't know what to do with the summation. I could evaluate

$$ \int_{0}^{\infty} x^3 e^{-x} dx = 6$$

pleas help me to get the rest of it, I see the answer comes with $\pi$, how do I get it in this kind of equation? Is there any special trick to solve these type of integrals?
thanks in advance.

 

[ShayanJ]

Under the condition that both the series and the integral converge, you can swap them. Here we're sure that the series converges because we got it by taylor-expanding a function. About the integral, we have a polynomial against an exponential with negative exponent which means converges too. So we have $\displaystyle \sum_{n=0}^\infty \int_0^\infty x^3 e^{-x}e^{-nx} dx=\sum_{n=0}^\infty \int_0^\infty x^3 e^{-(n+1)x}$. Now you if you integrate by parts three times(each time considering u to be the algebraic factor and dv to be the exponential one), you can obtain the solution to the integral.

 

[azoroth134]

Ah great, thank you very much, I almost got it on the track now, this is

$$ \int_{0}^{\infty} x^3 e^{-(n+1)x} dx = 6 (n+1)^{-4}$$

can you just tell me a little about how to evaluate the summation of $$\sum \frac{1}{(n+1)^4}$$ though, I think I forgot this evaluation.

thanks in advance

 

[ShayanJ]

The series $\displaystyle \sum_{n=0}^\infty \frac{1}{(n+1)^4}$ can be written as $\displaystyle \sum_{n=1}^\infty \frac{1}{n^4}$ which is equal to $\zeta(4)$, where $\zeta(s)$ is the Riemann zeta function. So you don't need to evaluate the series, just look for tables of the values for this function or compute it using math softwares.

 

[azoroth134]

problem solved, thanks a lot