Giải thích các bước giải:
Ta có :
$A=\dfrac{1}{2}+\dfrac{2}{2^2}+\dfrac{3}{2^3}+..+\dfrac{100}{2^{100}}$
$\to 2A=1+\dfrac{2}{2}+\dfrac{3}{2^2}+..+\dfrac{100}{2^{99}}$
$\to 2A-A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+..+\dfrac{1}{2^{99}}-\dfrac{100}{2^{100}}$
$\to A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+..+\dfrac{1}{2^{99}}-\dfrac{100}{2^{100}}$
$\to 2A=2+1+\dfrac{1}{2}+..+\dfrac{1}{2^{98}}-\dfrac{100}{2^{99}}$
$\to 2A-A=2-\dfrac{100}{2^{99}}-\dfrac{1}{2^{99}}+\dfrac{100}{2^{100}}$
$\to A=2-(\dfrac{100}{2^{99}}+\dfrac{1}{2^{99}})+\dfrac{50}{2^{99}}$
$\to A=2-\dfrac{101}{2^{99}}+\dfrac{50}{2^{99}}$
$\to A=2-\dfrac{101-50}{2^{99}}$
$\to A=2-\dfrac{51}{2^{99}}$
$\to 2-\dfrac{1}{1000}<A<2$
$\to 1,999<A<2$
