Cho A= 1 + 1/2 + 1/3 + 1/4 +…+ 1/2^100 – 1. Chứng minh 50 < A

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1. `A=1+1/2+1/3+1/4+…+1/(2^(100)-1)`

   `CM:50<A`

   `+)` Ta có :

   `A=1+(1/2+1/3)+(1/4+1/5+1/6+1/7)+(1/8+1/9+..+1/15)+..+(1/(2^99)+1/(2^(99)+1)+..+1/(2^(100)-1)`

   `99` nhóm `<1+2.1/2+2^(2).1/(2^3).1/(2^3)+…+2^(99).1/(2^99))`

   `<=>1+1+1+…+1=100`

   `<=>A1+1/2+2.1/(2(2^2))+2^2.1/(2^3).1/(2^4)+..+(1^991)/(2^100)-1-1/(2^100)=1+1/2+1/2+…+1/2-1/(2^100)`

   `<=>1+100.12-1/(2^100)`

   `<=>50<50+1-1/(2^100)`

   Vậy `50<A`

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