[tex]\mathbf{S=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2} +...+\dfrac{1}{2011^2} <\dfrac{2010}{2011} }[/tex]
[tex]\mathbf{\implies \dfrac{1}{2^2} <\dfrac{1}{1\cdot 2} }[/tex]
[tex]\mathbf{\implies \dfrac{1}{3^2} <\dfrac{1}{2 \cdot 3} }[/tex]
[tex]\mathbf{\implies \dfrac{1}{4^2} <\dfrac{1}{3 \cdot 4} }[/tex]
[tex]\mathbf{\implies \dfrac{1}{2011^2} <\dfrac{1}{2010\cdot 2011} }[/tex]
[tex]\mathbf{\implies S<S_2}[/tex]
[tex]\mathbf{S_2=\dfrac{1}{1\cdot 2}+\dfrac{1}{2\cdot 3} +\dfrac{1}{3\cdot 4} +...+\dfrac{1}{2010\cdot 2011} }[/tex]
[tex]\mathbf{S_2=\dfrac{2-1}{1\cdot 2}+\dfrac{3-2}{2\cdot 3} +\dfrac{4-3}{3\cdot 4}+...+\dfrac{2011-2010}{2010 \cdot 2011} }[/tex]
[tex]\mathbf{S_2=\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3} -\dfrac{1}{4} +...+\dfrac{1}{2010} -\dfrac{1}{2011} }[/tex]
[tex]\mathbf{ Observam \: ca \: toate \: fractiile \:se \: simplifica\: cu \:exceptia\:primei \: si \: a \: ultimei}[/tex]
[tex]\mathbf{\implies S_2= \dfrac{1}{1} -\dfrac{1}{2011}=\dfrac{2011}{2011}-\dfrac{1}{2011} =\dfrac{2010}{2011} }[/tex]
[tex]\mathbf{Stiind \:\; ca \:\; S<S_2 \implies S<\dfrac{2010}{2011} }[/tex]
[tex]{\boxed{\mathbf{\implies \dfrac{1}{2^2}+\dfrac{1}{3^2} +\frac{1}{4^2}+...+\frac{1}{2011^2 } <\dfrac{2010}{2011} }}}[/tex]
