CMR : 1/1^2 + 1/2^2 + 1/3^2 + …..+1/n^2 < 2 mọi số nguyên dương n

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:

\(\begin{array}{l}
\frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + …. + \frac{1}{{{n^2}}}\\
 < 1 + \frac{1}{{1.2}} + \frac{1}{{2.3}} + \frac{1}{{3.4}} + …. + \frac{1}{{\left( {n – 1} \right).n}}\\
 = 1 + 1 – \frac{1}{2} + \frac{1}{2} – \frac{1}{3} + \frac{1}{3} – \frac{1}{4} + …. + \frac{1}{{n – 1}} – \frac{1}{n}\\
 = 2 – \frac{1}{n} < 2,\,\,\,\,\forall n
\end{array}\)