Question

A = \(\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+...+\frac{1}{2007\cdot2009}+\frac{1}{2009\cdot2011}\)=?

Answer 1

\(2.A=\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+....+\frac{2}{2007.2009}+\frac{2}{2009.2011}\)

\(2.A=\frac{3-1}{1.3}+\frac{5-3}{3.5}+\frac{7-5}{5.7}+...+\frac{2009-2007}{2007.2009}+\frac{2011-2009}{2009.2011}\)

\(2.A=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2007}-\frac{1}{2009}+\frac{1}{2009}-\frac{1}{2011}=1-\frac{1}{2011}=\frac{2010}{2011}\)

=>A = \(\frac{2010}{2011}:2=\frac{1005}{2011}\)