Question

chứng tỏ rằng:1/5 + 1/13 +1/25 +...+1/n²+(n+1)²  <  1/2 với mọi n thuộc N

Answer 1

Ta có: \(n^2+\left(n+1\right)^2>2n\left(n+1\right)\)

\(\Rightarrow\frac{1}{5}+\frac{1}{13}+...+\frac{1}{n^2+\left(n+1\right)^2}\)

\(=\frac{1}{1^2+2^2}+\frac{1}{2^2+3^2}+...+\frac{1}{n^2+\left(n+1\right)^2}< \frac{1}{2.1.2}+\frac{1}{2.2.3}+...+\frac{1}{2.n.\left(n+1\right)}\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{n.\left(n+1\right)}\right)\)

\(=\frac{1}{2}.\left(1-\frac{1}{n+1}\right)< \frac{1}{2}\)