Cho n là số nguyên dương. Chứng minh:
1 + $\frac{1}{2^{2}}+\frac{1}{3^{2}}+...+\frac{1}{n^{2}}$ <$\frac{5}{3}$
1 + $\frac{1}{2^{2}}+\frac{1}{3^{2}}+...+\frac{1}{n^{2}}$ <$\frac{5}{3}$
Started By volehoangdck269, 04-05-2016 - 21:38
#1
Posted 04-05-2016 - 21:38
#2
Posted 05-05-2016 - 16:05
Cho n là số nguyên dương. Chứng minh:
1 + $\frac{1}{2^{2}}+\frac{1}{3^{2}}+...+\frac{1}{n^{2}}$ <$\frac{5}{3}$
Với mọi $k\geq 1$ thì $\frac{1}{k^2}= \frac{4}{4k^2}< \frac{4}{4k^2-1}= 2(\frac{1}{2k-1}-\frac{1}{2k+1})$
Nên $1+\frac{1}{2^2}+...+\frac{1}{n^2}< 1+\frac{2}{3}-\frac{2}{2n+1}< \frac{5}{3}$
P/s: bài toán mạnh hơn: Cho n là số nguyên dương. Chứng minh rằng $1+\frac{1}{2^2}+...+\frac{1}{n^2}< \frac{79}{48}$
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