Cho biểu thức A = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... + 1/49 - 1/50. Chứng minh A < 5/6
Giải thích
\[A = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + ... + \frac{1}{{49}} - \frac{1}{{50}}\]
\[ = \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{{49}} - \frac{1}{{50}}\]
\[ = \frac{5}{6} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + ... + \frac{1}{{49}} - \frac{1}{{50}}\]
\[ = \frac{5}{6} + \left( {\frac{1}{5} - \frac{1}{4}} \right) + \left( {\frac{1}{7} - \frac{1}{6}} \right) + ... + \left( {\frac{1}{{49}} - \frac{1}{{48}}} \right) + \left( { - \frac{1}{{50}}} \right)\]
Mà \[\frac{1}{5} - \frac{1}{4} < 0\]
\[\frac{1}{7} - \frac{1}{6} < 0\]
...
\[\frac{1}{{49}} - \frac{1}{{48}} < 0\]
\[ - \frac{1}{{50}} < 0\]
Do đó \[A < \frac{5}{6}.\]