Question

Найдите lim(n→∞)(1/(1⋅2)+1/(2⋅3)+1/(3⋅4)+…+1/(n⋅(n+1))

Answer 1

$\lim\limits_{n\to\infty}\left(\dfrac1{1\cdot2}+\dfrac1{2\cdot3}+\dots+\dfrac1{n(n+1)}\right)=\lim\limits_{n\to\infty}\left(\dfrac11-\dfrac12+\dfrac12-\dfrac13+\dots\right.\\\left.+\dfrac1n-\dfrac1{n+1}\right)=\lim\limits_{n\to\infty}\left(1-\dfrac1{n+1}\right)=1$