wy
Lv 7
wy 發問於 科學及數學數學 · 5 年前

Math. Induction?

Prove by M. I. that

1 - 1/2 + 1/3 - 1/4 + ......+ 1/(2n + 1) - 1/(2n + 2) = 1/(n + 2) + 1/(n + 3) + ..... + 1/(2n + 2)

1 個解答

評分
  • 5 年前

    Prove by M. I. that

    1-1/2+1/3-1/4+ ......+1/(2n+1)-1/(2n + 2)=1/(n + 2)+1/(n + 3)+ ..... +1/(2n + 2)

    Sol

    當n=1時

    左=1/1-1/2+1/3-1/4=(12-6+4-3)/12=7/12

    右=1/3+1/4=7/12

    左=右

    n=1時為真……………………….

    設 n=k時為真即

    1-1/2+1/3-1/4+ ......+1/(2k+1)-1/(2k + 2)=1/(k + 2)+1/(k + 3)+ ..... +1/(2k + 2)

    [1-1/2+1/3-1/4+ ......+1/(2k+1)-1/(2k + 2)]-1/(2k+2+2)

    =[1/(k + 2)+1/(k + 3)+ ..... +1/(2k + 2)]-1/(2k+4)

    =[1/(k + 3)+ ..... +1/(2k + 2)]+1/(k+2)-1/(2k+4)

    =[1/(k + 3)+ ..... +1/(2k + 2)]+2/(2k+4)-1/(2k+4)

    =[1/(k + 3)+ ..... +1/(2k + 2)]+1/(2k+4)

    =1/(k + 3)+ ..... +1/(2k + 2)+1/(2k+4)

    =1/[(k+1)+2)+ ..... +1/[2(k+1)+1/[2(k+1)+2]

    So

    n=k+1時為真

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