计算积分
[int_{0}^{1}xleft[frac{1}{x} ight]dx=frac{pi^{2}}{12}]
解:
egin{align*}int_{0}^{1}xleft[frac{1}{x} ight]dx&=-sum_{n=1}^{infty}int_{frac{1}{n}}^{frac{1}{n+1}}xleft[frac{1}{x} ight]dx \&=sum_{n=1}^{infty}frac{n}{2}left[frac{1}{n^{2}}-frac{1}{(n+1)^{2}} ight]\&=frac{1}{2} sum_{n=1}^{infty}(frac{1}{n}-frac{1}{n+1}+frac{1}{(n+1)^{2}})\&=frac{pi ^{2}}{12}end{align*}
对于 $s>1$,同样的方法可以计算
$$zeta(s)=sint_{1}^{infty} frac{[ x ]}{x^{s+1}} dx$$