求f(m)=Sigma from k=1 to infinity ((k-1)!)/((k+m)!)

求f(m)=Sigma from k=1 to infinity ((k-1)!)/((k+m)!)
如下例子：
f(1)=sigma from k=1 to infinity 1/(k*(k+1))=1
f(2)=sigma from k=1 to infinity 1/(k*(k+1)*(k+2))=1/4
f(3)=sigma from k=1 to infinity 1/(k*(k+1)*(k+2)*(k+3))=1/96
求：
推广到m的表达式

或者针对某个特定m的证明过程
thxs...