Chứng minh B = 1/2^2 + 1/3^2 + 1/4^2 + 1/5^2 + 1/6^2 + 1/7^2 + 1/8^2 < 1
Chứng minh \[B = \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + \frac{1}{{{5^2}}} + \frac{1}{{{6^2}}} + \frac{1}{{{7^2}}} + \frac{1}{{{8^2}}} < 1\].
Dễ thấy
\[\frac{1}{{{2^2}}} = \frac{1}{{2.2}} < \frac{1}{{1.2}}\].
\[\frac{1}{{{3^2}}} = \frac{1}{{3.3}} < \frac{1}{{2.3}}\]
…………….
\[\frac{1}{{{8^2}}} = \frac{1}{{8.8}} < \frac{1}{{7.8}}\]
Cộng vế theo vế ta được:
\[B < \frac{1}{{1.2}} + \frac{1}{{2.3}} + \frac{1}{{3.4}} + ... + \frac{1}{{7.8}}\]
\[ = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + ... + \frac{1}{7} - \frac{1}{8}\]
\[ = 1 - \frac{1}{8} = \frac{7}{8} < 1\]
Vậy \[B = \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + ... + \frac{1}{{{8^2}}} < 1\] (đpcm)