\(\frac{1}{1\times3}+\frac{1}{3\times5}+\frac{1}{5\times7}+...+\frac{1}{2009\times2011}\)
= \(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2009}-\frac{1}{2011}\)
= \(1-\frac{1}{2011}\)
= \(\frac{2010}{2011}\)
Đặt A=1/1.3+1/3.5+1/5.7+...+1/2009.2011
2A=2/1.3+2/3.5+2/5.7+...+2/2009.2011
2A=1/1-1/3+1/3-1/5+1/5-1/7+...+1/2009-1/2011
2A=1-1/2011=2011/2011-1/2011=2010/2011
A=2010/2011.1/2=1005/2011
Đặt A=1/1.3+1/3.5+1/5.7+...+1/2009.2011
2A=2/1.3+2/3.5+2/5.7+...+2/2009.2011
2A=1/1-1/3+1/3-1/5+1/5-1/7+...+1/2009-1/2011
2A=1-1/2011=2011/2011-1/2011=2010/2011
A=2010/2011:2
=1005/2011
