Cho ​\(S = \frac{1}{{{5^2}}} + \frac{1}{{{7^2}}} + \frac{1}{{{9^2}}} + .... + \frac{1}{{{{103}^2}}}\). Kết luận nào sau đây đúng?

* Hướng dẫn giải

Ta có:

\(\begin{array}{l} \frac{1}{{{5^2}}} < \frac{1}{{4.6}};\frac{1}{{{5^7}}} < \frac{1}{{6.8}};\frac{1}{{{9^2}}} < \frac{1}{{8.10}};.......;\frac{1}{{{{103}^2}}} < \frac{1}{{102.104}}\\ \Rightarrow S = \frac{1}{{{5^2}}} + \frac{1}{{{7^2}}} + \frac{1}{{{9^2}}} + .... + \frac{1}{{{{103}^2}}} < \frac{1}{{4.6}} + \frac{1}{{6.8}} + \frac{1}{{8.10}} + .... + \frac{1}{{102.104}}\\ \Rightarrow S < \frac{1}{2}\left( {\frac{1}{4} - \frac{1}{6} + \frac{1}{6} - \frac{1}{8} + \frac{1}{8} - \frac{1}{{10}} + ..... + \frac{1}{{102}} - \frac{1}{{104}}} \right)\\ \Rightarrow S < \frac{1}{2}\left( {\frac{1}{4} - \frac{1}{{104}}} \right)\\ \Rightarrow S < \frac{{25}}{{208}} < \frac{{25}}{{160}}\\ \Rightarrow S < \frac{5}{{32}} \end{array}\)