1/(1*2)+2/(1*2*3)+3/(1*2*3*4)+4/(1*2*3*4*5)+、、、+9/(1*2*3*4*5

1个回答

  • n!=1*2*3*…*n

    1/(1*2)=(2-1)/(1*2)=2/(1*2)-1/(1*2)=1-1/(1*2)=1-1/(2!)

    2/(1*2*3)=(3-1)/(1*2*3)=3/(1*2*3)-1/(1*2*3)=1/(1*2)-1/(1*2*3)=1/(2!)-1/(3!)

    同理 3/(1*2*3*4)=1/(1*2*3)-1/(1*2*3*4)=1/(3!)-1/(4!)

    …………………

    9/(10!)=1/(1*2*3*4*5*6*7*8*9)-1/(1*2*3*4*5*6*7*8*9*10)=1/(9!)-1/(10!)

    所以原式=1-1/(2!)+1/(!2)-1/(3!)+1/(3!)-1/(4!)+……+1/(9!)-1/(10!)

    =1-(1/10!)