Lim n-8 (1/1*2+1/2*3+...+1/(n-1)n

Решите задачу:

$\displaystyle \lim_{n \to \infty} \bigg( \frac{1}{1\cdot2}+ \frac{1}{2\cdot3}+ \frac{1}{3\cdot4}+...+ \frac{1}{(n-1)\cdot n} \bigg)=\\ \\ \\ =\lim_{n \to \infty} \bigg( \frac{2-1}{1\cdot2}+ \frac{3-2}{2\cdot3}+ \frac{4-3}{3\cdot4}+...+ \frac{n-(n-1)}{(n-1)\cdot n}\bigg)=\\ \\ \\ =\lim_{n \to \infty} \bigg(1- \frac{1}{2}+ \frac{1}{2}- \frac{1}{3}+ \frac{1}{3}- \frac{1}{4} +...+ \frac{1}{n-1}- \frac{1}{n} \bigg)=\\ \\ \\ =\lim_{n \to \infty} \bigg(1- \frac{1}{n}\bigg)=1-0=1$