Question

Cho số n lớn hơn hoạc bằng 2 (n thuộc N). So sánh:

A= \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}và1\)

Answer 1

Ta có : 

\(\frac{1}{2^2}=\frac{1}{2.2}<\frac{1}{1.2}\)

\(\frac{1}{3^2}=\frac{1}{3.3}<\frac{1}{2.3}\)

....

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

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

Mà \(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{\left(n-1\right).n}<1\)nên \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}<1\)