Question

lim  (1/1*2+1/2*3+...+1/(n-1)n

Answer 1

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