\(S=\dfrac{1^2}{1}-\dfrac{1}{1}+\dfrac{2^2}{2^2}-\dfrac{1}{2^2}+...+\dfrac{n^2}{n^2}-\dfrac{1}{n^2}\)
\(S=1-\dfrac{1}{1}+1-\dfrac{1}{2^2}+...+1-\dfrac{1}{n^2}\)
\(S=n-\left(\dfrac{1}{1}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}\right)=n-A\)
Xét \(A=\dfrac{1}{1}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}=1+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}>1\) (1)
\(A=\dfrac{1}{1}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< \dfrac{1}{1}+\dfrac{1}{1.2}+...+\dfrac{1}{\left(n-1\right)n}\)
\(\Rightarrow A< \dfrac{1}{1}+\dfrac{1}{1}-\dfrac{1}{2}+...+\dfrac{1}{n-1}-\dfrac{1}{n}=2-\dfrac{1}{n}< 2\) (2)
Từ (1),(2) \(\Rightarrow1< A< 2\Rightarrow A\) nằm giữa 2 số nguyên liên tiếp nên A không phải là số nguyên.
Mà \(S=n-A\), do \(n\) nguyên, \(A\) không nguyên \(\Rightarrow S\) không nguyên