Question

calculati cu rez completa va rogg

Answer 1

$\frac{1}{ \sqrt{2} +1} + \frac{1}{ \sqrt{3} + \sqrt{2} } +\frac{1}{ \sqrt{4} + \sqrt{3} } +...+\frac{1}{ \sqrt{2011} + \sqrt{2010} } = \\ \frac{\sqrt{2} -1}{ (\sqrt{2} +1)(\sqrt{2} -1)} + \frac{\sqrt{3} - \sqrt{2}}{ (\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) } +\frac{\sqrt{4} - \sqrt{3}}{ (\sqrt{4} + \sqrt{3})(\sqrt{4} - \sqrt{3}) } +...+ \\ +\frac{\sqrt{2011} - \sqrt{2010}}{ (\sqrt{2011} + \sqrt{2010})(\sqrt{2011} - \sqrt{2010}) }=$
$\frac{\sqrt{2} -1}{ 2-1} + \frac{\sqrt{3} - \sqrt{2}}{ 3-2 } +\frac{\sqrt{4} - \sqrt{3}}{ 4-3} +...+ \frac{\sqrt{2011} - \sqrt{2010}}{ 2011-2010}= \\ \frac{\sqrt{2} -1}{ 1} + \frac{\sqrt{3} - \sqrt{2}}{1 } +\frac{\sqrt{4} - \sqrt{3}}{1} +...+ \frac{\sqrt{2011} - \sqrt{2010}}{ 1}= \\ \sqrt{2} -1+ \sqrt{3}- \sqrt{2} +\sqrt{4}- \sqrt{3} +...+\sqrt{2011}- \sqrt{2010}= \\ \sqrt{2011}-1$