Question

Va rog, dau coroana!

Answer 1

b) 1/1*2 + 1/2*3 + 1/3*4 +........... +1 /2013*2014 =
= 1/1 - 1/2 + 1/2 -1/3 +1/3 -1/4 +1/4 -1/5 +............+ 1/2013 - 1/2014 =
=1 - 1/2014 =  2013/2014

Answer 2

$\displaystyle a). \frac{1}{n} - \frac{1}{n+m} = \frac{n+m}{n(n+m)} - \frac{n}{n(n+m)} = \frac{n+m-n}{n(n+m)} = \frac{m}{n(n+m)} \\ \\ b). \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} +...+ \frac{1}{2013 \cdot 2014} = \\ \\ = \frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} +...+ \frac{1}{2013} - \frac{1}{2014} = \\ \\ = {^{2014)} 1- \frac{1}{2014} = \frac{2014-1}{2014} = \frac{2013}{2014}$