Question

A= 1+\(\frac{2^2}{3^2}\)+\(\frac{2^2}{5^2}\)+\(\frac{2^2}{7^2}\)+...+\(\frac{2^2}{2009^2}\)So sánh Avới 3

 

Answer 1

Ta có:

\(\frac{2^2}{3^2}>\frac{2^2}{2009^2}\)

\(\frac{2^2}{5^2}>\frac{2^2}{2009^2}\)

\(\frac{2^2}{7^2}>\frac{2^2}{2009^2}\)

        .........

\(\frac{2^2}{2009^2}=\frac{2^2}{2009^2}\)

\(\Rightarrow\frac{2^2}{3^2}+\frac{2^2}{5^2}+\frac{2^2}{7^2}+...+\frac{2^2}{2009^2}>\frac{2^2}{2009^2}+\frac{2^2}{2009^2}+\frac{2^2}{2009^2}+...+\frac{2^2}{2009^2}=\frac{2^2.1004}{2009^2}=\frac{4016}{2009^2}\)(1004 phân số \(\frac{2^2}{2009^2}\)) . Mà:

\(\frac{4016}{2009^2}< 3\)

=> A < 3