Question

CMR:Với mọi số tự nhiên n>2 hãy so sánh

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

Answer 1

Ta có :

\(\frac{1}{2^2}< \frac{1}{1.2}\)

\(\frac{1}{3^2}< \frac{1}{2.3}\)

...

\(\frac{1}{n^2}< \frac{1}{\left(n-1\right)n}\)

\(\Rightarrow A=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< \frac{1}{1.2}+\frac{1}{2.3}+..+\frac{1}{\left(n-1\right)n}\)\(\Rightarrow A< \frac{1}{1.2}+\frac{1}{2.3}+..+\frac{1}{\left(n-1\right)n}\)

\(\Rightarrow A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{n-1}-\frac{1}{n}\)

\(\Rightarrow A< 1-\frac{1}{n}< 1\left(đpcm\right)\)