Question

Заметив, что 1/(k(k + 1)) = 1/k - 1/(k + 1) вычислите 1/(1 * 2) + 1/(2 * 3) + 1/(3 * 1) +...+ 1 2014*2015 .​

Answer 1

$\displaystyle \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{2014\cdot2015}=\bigg(1-\frac{1}{2}\bigg)+\bigg(\frac{1}{2}-\frac{1}{3}\bigg)+\bigg(\frac{1}{3}-\frac{1}{4}\bigg)+...\\\\\\+\bigg(\frac{1}{2013}-\frac{1}{2014}\bigg)+\bigg(\frac{1}{2014}-\frac{1}{2015}\bigg)=1-\frac{1}{2015}=\frac{2014}{2015}$