1.2-1/2! + 2.3-1/3! + 3.4/4! + … + 99.100-1/100! < 2

0 bình luận về “1.2-1/2! + 2.3-1/3! + 3.4/4! + … + 99.100-1/100! < 2”

1. `(1.2 -1)/(2!) +(2.3 -1)/(3!)+(3.4)/(4!)+…+ (99.100-1)/(100!)`

   Đặt `A=(1.2 -1)/(2!) +(2.3 -1)/(3!)+(3.4)/(4!)+…+ (99.100-1)/(100!)`

   `=>A=(1.2)/(2!)- 1/(2!)+ (2.3)/(3!)-1/(3!)+…+(99.100)/(100!)-1/(100!)`

   `=>A=[((1.2)/(2!)+(2.3)/(3!)+….+(99.100)/(100!))-(1/(2!)+1/(3!)+…+1/(100!)]`

   `=>A=[(1+1+1/(2!)+….+1/(98!))-(1/(2!)+1/(3!)+…+1/(100!)]`

   `=>A=1-1/(99!)-1/(100!)`

   Mà `1-1/(99!)-1/(100!)<1=>1-1/(99!)-1/(100!)<2` `->đpcm`

    

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2. `A=1×2−1/2!+2×3−1/3!+3×4−1/4!+…+99×100−1/100!`

   `=1×2/2!−1/2!+2×3/3!−1/3!+…+99×100/100!−1/100!`

   `=(1×2/2!+2×3/3!+3×4/4!+…+99×100/100!)−(1/2!+1/3!+1/4!+…+1/100!)`

   `=(1+1+1/2!+…+1/98!)−(1/2!+1/3!+1/4!+…+1/100!)`

   `=1+1−1/99!−1/100!`

   `=2−1/99!−1/100!`

   $Mà:$ `2−199!−1100!<2`

   `⇒A<2`

   $#JAW#$

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