\(C=\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}\)
\(\Rightarrow C>\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+...+\dfrac{1}{n\times\left(n+1\right)}\)
\(\Rightarrow C>\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n}-\dfrac{1}{n+1}\)
\(\Rightarrow C>1-\dfrac{1}{n+1}\)
\(\Rightarrow C>1\) (1)
Mặt khác:
\(C< 1+\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+...+\dfrac{1}{\left(n-1\right)\times n}\)
\(\Rightarrow C< 1+\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{\left(n-1\right)}-\dfrac{1}{n}\)
\(\Rightarrow C< 1+1-\dfrac{1}{n}\)
\(\Rightarrow C< 2-\dfrac{1}{n}\)
\(\Rightarrow C< 2\) (2)
Từ (1) và (2)=> 1<C<2
Vậy 1<C<2.(ĐPCM)
