Ta có :
\(B=\frac{2008}{1}+\frac{2007}{2}+...+\frac{2}{2007}+\frac{1}{2008}\)
\(\Rightarrow B=1+\left(\frac{2007}{2}+1\right)+...+\left(\frac{2}{2007}+1\right)+\left(\frac{1}{2008}+1\right)\)
\(\Rightarrow B=\frac{2009}{2009}+\frac{2009}{2}+...+\frac{2009}{2007}+\frac{2009}{2008}\)
\(\Rightarrow B=\frac{2009}{2}+...+\frac{2009}{2007}+\frac{2009}{2008}+\frac{2009}{2009}\)
\(\Rightarrow B=2009.\left(\frac{1}{2}+...+\frac{1}{2007}+\frac{1}{2008}+\frac{1}{2009}\right)\)
\(\Rightarrow B=2009.A\)
\(\Rightarrow\frac{A}{B}=\frac{A}{2009.A}=\frac{1}{2009}\)
Chúc bạn học tốt !!!
\(B=\frac{2008}{1}+\frac{2007}{2}+...+\frac{2}{2007}+\frac{1}{2008}\)
\(=2008+\left(\frac{2007}{2}+1\right)+\left(\frac{2006}{3}+1\right)+...+\left(\frac{2}{2007}+1\right)+\left(\frac{1}{2008}+1\right)-2007\)
\(=\frac{2009}{2}+\frac{2009}{3}+...+\frac{2009}{2007}+\frac{2009}{2008}+1\)
\(=\frac{2009}{2}+\frac{2009}{3}+...+\frac{2009}{2007}+\frac{2009}{2008}+\frac{2009}{2009}\)
\(=2009\cdot\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2009}\right)\)
=> \(\frac{A}{B}=\frac{\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2008}+\frac{1}{2009}}{2009\cdot\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2008}+\frac{1}{2009}\right)}=\frac{1}{2009}\)
