egin{align*}sum_{n=0}^{infty}frac{(n!)^{2}2^{n+1}}{(2n+1)!}&=sum_{n=0}^{infty}int_{0}^{1}t^{n}(1-t)^{n}2^{n+1}dt\&=2int_{0}^{1}sum_{0}^{infty}t^{n}(1-t)^{n}2^{n}dt\&=int_{0}^{1}frac{1}{(t-frac{1}{2})+frac{1}{4}}dt\&=2 arctan2(t-frac{1}{2})|_{0}^{1}\&=piend{align*}
特为http://www.cnblogs.com/zhangwenbiao/p/3705281.html写的代码.