Chứng minh 1/2^2+1/3^2+1/4^2+….+1/n^2<1

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1. Đáp án:

   $\begin{array}{l}
   A = \dfrac{1}{{{2^2}}} + \dfrac{1}{{{3^2}}} + \dfrac{1}{{{4^2}}} + … + \dfrac{1}{{{n^2}}}\\
   Do:1.2 < {2^2}\\
    \Rightarrow \dfrac{1}{{{2^2}}} < \dfrac{1}{{1.2}}\\
   TT:\dfrac{1}{{{3^2}}} < \dfrac{1}{{2.3}};\dfrac{1}{{{4^2}}} < \dfrac{1}{{3.4}};…;\dfrac{1}{{{n^2}}} < \dfrac{1}{{\left( {n – 1} \right).n}}\\
    \Rightarrow A < \dfrac{1}{{1.2}} + \dfrac{1}{{2.3}} + \dfrac{1}{{3.4}} + … + \dfrac{1}{{\left( {n – 1} \right).n}}\\
    \Rightarrow A < 1 – \dfrac{1}{2} + \dfrac{1}{2} – \dfrac{1}{3} + \dfrac{1}{3} – \dfrac{1}{4} + .. + \dfrac{1}{{n – 1}} – \dfrac{1}{n}\\
    \Rightarrow A < 1 – \dfrac{1}{n} < 1\\
   Vậy\,A < 1
   \end{array}$

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