[tex]\displaystyle\\
\text{Folosim formula:}\\\\
\frac{2}{n(n+2)}= \frac{1}{n} - \frac{1}{n+2} \\\\
\left( \frac{1}{1\times3}+\frac{1}{3\times5}+\frac{1}{5\times7}+\cdots+\frac{1}{2009\times2011} \right)\times4022=\\\\
=\left(\frac{1}{1\times3}+\frac{1}{3\times5}+\frac{1}{5\times7}+\cdots+\frac{1}{2009\times2011}\right)\times2\times2011=\\\\
=\left(\frac{2}{1\times3}+\frac{2}{3\times5}+\frac{2}{5\times7}+\cdots+\frac{2}{2009\times2011} \right)\times2011=
[/tex]
[tex]\displaystyle\\
=\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\cdots+\frac{1}{2009}-\frac{1}{2011} \right)\times2011=\\\\
\text{Observam ca fractiile se reduc 2 cate 2 si raman prima si ultima.}\\\\
=\left(\frac{1}{1}-\frac{1}{2011} \right)\times2011=\\\\
=\left(\frac{2011}{2011}-\frac{1}{2011} \right)\times2011=\frac{2010}{2011}\times 2011 = \boxed{\bf 2010}[/tex]
