Question

СРОЧНО НАДО!!!!!!!!!!!
а=1/корень1+ корень 2 +1/ корень 2+ корень 3 +1 / корень 3+ корень 4 + ......1/ корень 2010+ корень 2011


В числителе везде 1, а в знаменателе по два корня..например корень1+ корень 2

Answer 1

$A = sumlimits_{n = 1}^{2010} frac{1}{sqrt{n}+sqrt{n+1}} = frac{1}{sqrt{1}+sqrt{2}} + frac{1}{sqrt{2}+sqrt{3}} + ... + frac{1}{sqrt{2010}+sqrt{2011}} = \\ frac{(sqrt{2} - sqrt{1})}{(sqrt{1}+sqrt{2})(sqrt{2} - sqrt{1})} + frac{(sqrt{3} - sqrt{2})}{(sqrt{2}+sqrt{3})(sqrt{3} - sqrt{2})} + ... + frac{(sqrt{2011} - sqrt{2010})}{(sqrt{2010}+sqrt{2011})(sqrt{2011} - sqrt{2010})} =$

$frac{sqrt{2} - sqrt{1}}{2 - 1} + frac{sqrt{3} - sqrt{2}}{3 -2} + ... + frac{sqrt{2011} - sqrt{2010}}{2011 - 2010} = \\ sqrt{2} - sqrt{1} + sqrt{3} - sqrt{2} + ... + sqrt{2010} -sqrt{2009} + sqrt{2011} - sqrt{2010} =\\ boxed{sqrt{2011} - 1}$