chứng tỏ rằng 1+1/2+1/3+1/4+…+1/31+1/32>7/2 28/07/2021 Bởi Aubrey chứng tỏ rằng 1+1/2+1/3+1/4+…+1/31+1/32>7/2
$\begin{array}{l}\quad 1+\dfrac12+\dfrac13+\dfrac14+\ldots+\dfrac1{32}\\=1+\dfrac12+\left(\dfrac13+\dfrac14\right)+\underbrace{\left(\dfrac15+\ldots+\dfrac18\right)}_{\text{có 4 số}}+\underbrace{\left(\dfrac19+\ldots+\dfrac1{16}\right)}_{\text{có 8 số}}+\underbrace{\left(\dfrac1{17}+\ldots+\dfrac1{32}\right)}_{\text{có 16 số}}\\>1+\dfrac12+\left(\dfrac14+\dfrac14\right)+\underbrace{\left(\dfrac18+\ldots+\dfrac18\right)}_{\text{có 4 số}}+\underbrace{\left(\dfrac1{16}+\ldots+\dfrac1{16}\right)}_{\text{có 8 số}}+\underbrace{\left(\dfrac1{32}+\ldots+\dfrac1{32}\right)}_{\text{có 16 số}}\\=1+\dfrac12+\dfrac24+\dfrac48+\dfrac8{16}+\dfrac{16}{32}\\=\left(\dfrac12+\dfrac12\right)+\dfrac12+\dfrac12+\dfrac12+\dfrac12+\dfrac12\\=\dfrac72\\\to1+\dfrac12+\dfrac13+\dfrac14+\ldots+\dfrac1{32}>\dfrac72 \end{array}$ Bình luận
$\begin{array}{l}\quad 1+\dfrac12+\dfrac13+\dfrac14+\ldots+\dfrac1{32}\\=1+\dfrac12+\left(\dfrac13+\dfrac14\right)+\underbrace{\left(\dfrac15+\ldots+\dfrac18\right)}_{\text{có 4 số}}+\underbrace{\left(\dfrac19+\ldots+\dfrac1{16}\right)}_{\text{có 8 số}}+\underbrace{\left(\dfrac1{17}+\ldots+\dfrac1{32}\right)}_{\text{có 16 số}}\\>1+\dfrac12+\left(\dfrac14+\dfrac14\right)+\underbrace{\left(\dfrac18+\ldots+\dfrac18\right)}_{\text{có 4 số}}+\underbrace{\left(\dfrac1{16}+\ldots+\dfrac1{16}\right)}_{\text{có 8 số}}+\underbrace{\left(\dfrac1{32}+\ldots+\dfrac1{32}\right)}_{\text{có 16 số}}\\=1+\dfrac12+\dfrac24+\dfrac48+\dfrac8{16}+\dfrac{16}{32}\\=\left(\dfrac12+\dfrac12\right)+\dfrac12+\dfrac12+\dfrac12+\dfrac12+\dfrac12\\=\dfrac72\\\to1+\dfrac12+\dfrac13+\dfrac14+\ldots+\dfrac1{32}>\dfrac72 \end{array}$