Question

Найти предел:
$lim_{n to infty}(frac{1}{2*3}+frac{1}{3*4}+...+frac{1}{(n+1)(n+2)})$

Answer 1

$displaystyle lim_{n to infty}bigg( dfrac{1}{2cdot 3}+ frac{1}{3cdot 4}+...+ frac{1}{(n+1)(n+2)} bigg)=\ \ \ = lim_{n to infty} bigg( frac{3-2}{2cdot 3} + frac{4-3}{3cdot 4} +...+ frac{n+2-(n+1)}{(n+1)(n+2)} bigg)=\ \ \ = lim_{n to infty} bigg(frac{1}{2}-frac{1}{3} +frac{1}{3}-frac{1}{4} +...+ frac{1}{n+1} -frac{1}{n+2} bigg)=\ \ \ = lim_{n to infty} bigg(frac{1}2}- frac{1}{n+2}bigg)=frac{1}{2}-0=frac{1}2}$