Giải thích các bước giải:
TQ:
\(\begin{array}{l}
{n^2} > n\left( {n - 1} \right) \Rightarrow \frac{1}{{{n^2}}} < \frac{1}{{n\left( {n - 1} \right)}}\\
\frac{1}{{n\left( {n - 1} \right)}} = \frac{{n - \left( {n - 1} \right)}}{{n\left( {n - 1} \right)}} = \frac{1}{{n - 1}} - \frac{1}{n}
\end{array}\)
Ta có:
\(\begin{array}{l}
\frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + .... + \frac{1}{{{n^2}}}\\
< 1 + \frac{1}{{1.2}} + \frac{1}{{2.3}} + \frac{1}{{3.4}} + .... + \frac{1}{{\left( {n - 1} \right).n}}\\
= 1 + 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + .... + \frac{1}{{n - 1}} - \frac{1}{n}\\
= 2 - \frac{1}{n} < 2,\,\,\,\,\forall n
\end{array}\)
