Toán Lớp 9: Chứng minh rằng
$\frac{1}{1^{2}}$ +$\frac{1}{2^{2}}$ +………….+$\frac{1}{n^{2}}$ < $\frac{5}{3}$
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Toán Lớp 9: Chứng minh rằng
$\frac{1}{1^{2}}$ +$\frac{1}{2^{2}}$ +………….+$\frac{1}{n^{2}}$ < $\frac{5}{3}$
Comments ( 2 )
⇒ 1/k² < 1/[(k-1).(k+1)] = [1/(k-1) – 1/(k+1)]/2 (*)
Áp dụng (*), ta có:
1/1^2+1/2² + 1/3² + 1/4² + … + 1/n²
< 1/1^2+1/2² + 1/(2.4) + 1/(3.5) + … + 1/[(n-1).(n+1)]
=1/1^2 1/2² + [1/2 – 1/4 + 1/3 – 1/5 + … + 1/(n-1) – 1/(n+1)]/2
= 1/1^2+1/2² + [1/2 + 1/3 – 1/n – 1/(n+1)]/2
= 5/3 – [1/n + 1/(n+1)]/2 < 5/3