=\sum^{\infty}_{n=1}\int^{\infty}_{0}dxx^3e^{-nx}=\sum^{\infty}_{n=1}\int^{\infty}_{0}\frac{dt}{n}\frac{t^3}{n^3}e^{-t}=\sum^{\infty}_{n=1}\frac{1}{n^4}\int^{\infty}_0dte^{-t}t^3=\Gamma(4)\zeta(4)=6\frac{\pi^4}{90}=\frac{\pi^4}{15}