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