1/2!+2/3!+3/4!……+n-1/n!=(2-1)/2!+(3-1)/3!+(4-1)/4!+...+(n-1)/n!=[2/2!-1/2!]+[3/3!-1/3!]+[4/4!-1/4!]+...+[n/n!-1/n!]=1-1/2!+1/2!-1/3!+1/3!-1/4!+...+1/(n-1)!-1/n!=1-1/n!