Convergence of Series: Checking Equality by Changing Limits

[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..

 

[Office_Shredder]

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

 

[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.