Toán Lớp 8: Chứng minh rằng:
H=1/2^2+1/3^2+1/4^2+..,+1/n^2<2/3 với n ∈N, n ≥2
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Toán Lớp 8: Chứng minh rằng:
H=1/2^2+1/3^2+1/4^2+..,+1/n^2<2/3 với n ∈N, n ≥2
Comments ( 1 )
$\text{Ta có H=$\frac{1}{2²}$ +$\frac{1}{3²}$ +………….+$\frac{1}{n²}$ }$
$\text{ ⇔H=$\frac{1}{2.2}$+$\frac{1}{3.3}$+………….+$\frac{1}{n.n}$ }$
$\text{<$\frac{1}{1.2}$+$\frac{1}{2.3}$+………..+$\frac{1}{n.(n-1)}$ }$
$\text{=$\frac{1}{1}$-$\frac{1}{2}$+$\frac{1}{2}$-$\frac{1}{3}$+……….+$\frac{1}{n-1}$-$\frac{1}{n}$ }$
$\text{=1-$\frac{1}{n}$ }$
$\text{≤1-$\frac{1}{2}$ (vì n≥2) }$
$\text{=$\frac{1}{2}$ }$
$\text{<$\frac{2}{3}$ }$
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