Giải thích các bước giải:
Ta có:
\[{n^2} > {n^2} - 1 \Rightarrow \frac{2}{{{n^2}}} < \frac{2}{{{n^2} - 1}} = \frac{2}{{\left( {n - 1} \right)\left( {n + 1} \right)}} = \frac{{\left( {n + 1} \right) - \left( {n - 1} \right)}}{{\left( {n - 1} \right)\left( {n + 1} \right)}} = \frac{1}{{n - 1}} - \frac{1}{{n + 1}}\]
Áp dụng ta có:
\[\begin{array}{l}
C = \frac{2}{{{3^2}}} + \frac{2}{{{5^2}}} + \frac{2}{{{7^2}}} + ... + \frac{2}{{{{2019}^2}}}\\
\Rightarrow C < \frac{1}{2} - \frac{1}{4} + \frac{1}{4} - \frac{1}{6} + \frac{1}{6} - \frac{1}{8} + .... + \frac{1}{{2018}} - \frac{1}{{2020}}\\
\Rightarrow C < \frac{1}{2} - \frac{1}{{2020}} = \frac{{2009}}{{2020}}
\end{array}\]
