# Are these two series equal?

1. Sep 23, 2009

### bob1182006

Are these two series equal? (Solved)

Not a homework problem but didn't think other forums were a better place to post this.

I'm trying to show:
$$\int_{0}^{\infty}\frac{x^3}{e^x-1}dx=\frac{\pi^4}{15}$$

After solving the integral, and checking it a few times, I get to this series:

$$\sum_{n=-1}^{\infty}\frac{6}{n^4}$$

I don't know if I can just say that:

$$\sum_{n=-1}^{\infty}\frac{1}{n^4}=\sum_{k=1}^{\infty}\frac{1}{k^4}$$

Since the RHS is equal to $\pi^4 / 90$

If I try to just pull out the first 2 terms on the LHS then I get the sum of 1+1/0, which is bad..

Last edited: Sep 23, 2009
2. Sep 23, 2009

### Office_Shredder

Staff Emeritus
You definitely can't say those series are equal. There is no mathematically sound way of calculation you could have gotten the LHS standing on its own as part of a solution, since it includes 1/0. There must be an error in your calculations

3. Sep 23, 2009

### bob1182006

Wow, thanks, just looked over what my professor did to start me off on this and noticed he made a mistake in a substitution and got n=-1 as the start of the series when it should have been n=1 so the series are equal!

Guess I should always look over everything, not just the part that I solved.