Question

найти предел (1/(2*4)+1/(4*6)+...+1/(2n*(2n+2))) СРОЧНО!

Answer 1

$displaystyle frac{1}{2cdot4}+ frac{1}{4cdot6}+...+frac{1}{2n cdot(2n+2)}= frac{1}{4} bigg( frac{1}{1cdot2}+frac{1}{2cdot3}+...+frac{1}{ncdot(n+1)}bigg)=\\ = frac{1}{4} bigg(frac{2-1}{1cdot2}+frac{3-2}{2cdot3}+...+frac{n+1-n}{ncdot(n+1)}bigg)=\ \ \ = frac{1}{4} bigg(1- frac{1}{2} + frac{1}{2} -frac{1}{3}+...+frac{1}{n} -frac{1}{n+1}bigg)= frac{1}{4} bigg(1-frac{1}{n+1}bigg)= frac{1}{4} ,~~ntoinfty$