so sánh 1/2+1/2^2+…+1/2^2014 và 1 so sánh 1+ 1/2^2+1/3^2+…+1/100^2<2

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1. Ta có:

   Đặt `A=1/2+1/2^2+…+1/2^2014`

   `2A=1+1/2+…+1/2^2013`
   `2A-A=(1+1/2+…+1/2^2013)-(1/2+1/2^2+…+1/2^2014)`
   `A=1-1/2^2014<1`
   Vậy `A<1`

   Ta có:

   Đặt `B=1/2^2+1/3^2+…+1/100^2`

   `C=1+ 1/2^2+1/3^2+…+1/100^2`

   Theo bài ra ta có:
   `1/2^2=1/2.2<1/1.2`
   `1/3^2=1/3.3<1/2.3`
   `…………….`
   `1/100^2=1/100.100<1/99.100`

   `=>B<1/1.2+1/2.3+…+1/99.100`
       `=1-1/2+1/2-1/3+…+1/99-1/100`
       `=1-1/100`
   `=>“C=1+1-1/100=2-1/100<2`
   Vậy `C<2`

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2. Đặt `:X= 1/2+1/2^2+…+1/2^(2014)`

   `⇔2X= 1+1/2^2+…+1/2^(2013)`

   `⇔2X-X=1+1/2^2+…+1/2^(2013)-(1/2+1/2^2+…+1/2^(2014))`

   `⇔X=1-1/(2^(2014))(ĐPCM)`

   `A=1+ 1/2^2+1/3^2+…+1/(100^2)≤1+1/(1.2)+1/(2.3)+…+1/(99.100)`

   `⇔A<1+1-1/2+1/2-1/3+…+1/(99)-1/(100)`

   `⇔A<2-1/(100)<2(ĐPCM)`

   `

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