CMR: \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}+\frac{1}{8^2}
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có: \(\frac{1}{2^2}< \frac{1}{1.2}\)(do 22>1.2)
\(\frac{1}{3^2}< \frac{1}{2.3}\)(do 32>2.3)
...........................................
\(\frac{1}{8^2}< \frac{1}{7.8}\)(do 82>7.8)
=> B < \(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{7.8}\)
=> B < \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{7}-\frac{1}{8}\)
=> B < \(1-\frac{1}{8}< 1\)
=> B < 1
Ta có \(\frac{1}{2^2}< \frac{1}{1.2}\)
\(\frac{1}{3^2}< \frac{1}{2.3}\)
....................
\(\frac{1}{8^2}< \frac{1}{7.8}\)
=> \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{8^2}< \frac{1}{1.2}+\frac{1}{2.3}+.......+\frac{1}{7.8}\)
=> \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{8^2}< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{7}-\frac{1}{8}\)
=> \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{8^2}< 1-\frac{1}{8}=\frac{7}{8}< 1\left(đpcm\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}\)
\(S=1-\frac{1}{9}=\frac{8}{9}\)
\(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{8^2}