由k/(k+1)!=(k+1-1)/(k+1)!=1/k!-1/(k+1)!,故得
1/2!+2/3!+3/4!+…+n/(n+1)!
=1/1!-1/2!+1/2!-1/3!+1/3!-1/4!+…+1/n!-1/(n+1)!
=1-1/(n+1)!