Trebuie sa ne folosim de o formula:
[tex] \frac{1}{a(a+k)}= \frac{1}{k} ( \frac{1}{a} - \frac{1}{a+k} ) [/tex]
[tex]S=(( \frac{1}{2}( \frac{1}{1} - \frac{1}{3} )+ \frac{1}{2}( \frac{1}{3} - \frac{1}{5} ))+...+ \frac{1}{2}( \frac{1}{2009} - \frac{1}{2011} ) )*4022\\\\
S=( \frac{1}{2}(1- \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} -...- \frac{1}{2009} + \frac{1}{2009}- \frac{1}{2011} ) )*4022[/tex]
Dupa cum vezi, se reduc termenii: 1/3 cu -1/3, 1/5 cu -1/5, si tot asa pana la 1/2009 cu -1/2009, si ne ramane 1 - 1/2011:
[tex]S= \frac{1}{2} (1- \frac{1}{2011} )*4022= \frac{2010*2011}{2011}=2010 [/tex]
