- #1
bob1182006
- 492
- 1
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:
[tex]\int_{0}^{\infty}\frac{x^3}{e^x-1}dx=\frac{\pi^4}{15}[/tex]
After solving the integral, and checking it a few times, I get to this series:
[tex]\sum_{n=-1}^{\infty}\frac{6}{n^4}[/tex]
I don't know if I can just say that:
[tex]\sum_{n=-1}^{\infty}\frac{1}{n^4}=\sum_{k=1}^{\infty}\frac{1}{k^4}[/tex]
Since the RHS is equal to [itex]\pi^4 / 90[/itex]
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..
Not a homework problem but didn't think other forums were a better place to post this.
I'm trying to show:
[tex]\int_{0}^{\infty}\frac{x^3}{e^x-1}dx=\frac{\pi^4}{15}[/tex]
After solving the integral, and checking it a few times, I get to this series:
[tex]\sum_{n=-1}^{\infty}\frac{6}{n^4}[/tex]
I don't know if I can just say that:
[tex]\sum_{n=-1}^{\infty}\frac{1}{n^4}=\sum_{k=1}^{\infty}\frac{1}{k^4}[/tex]
Since the RHS is equal to [itex]\pi^4 / 90[/itex]
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..
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