Question

1.Tìm số nguyên dương n thỏa mãn:

\(S=\sqrt{1+\frac{1}{1^2}+\frac{1}{^{2^2}}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{^{3^2}}}+...+\sqrt{1+\frac{1}{n^2}+\frac{1}{^{\left(n+1\right)^2}}}=\frac{2009^2-1}{2009}\)

2. Chứng minh rằng: với n là số nguyên dương bất kì thì:

\(A=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}<1.65\)

 

Answer 1

\(\sqrt{1+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}}=\sqrt{\left(1+\frac{1}{n}-\frac{1}{n+1}\right)^2}=1+\frac{1}{n}-\frac{1}{n+1}\)

\(S=1+1-\frac{1}{2}+1+\frac{1}{2}-\frac{1}{3}+....+1+\frac{1}{n}-\frac{1}{n+1}\)

   \(=n+1-\frac{1}{n+1}=\frac{\left(n+1\right)^2-1}{n+1}=\frac{2009^2-1}{2009}\Rightarrow n+1=2009\Rightarrow n=2008\)