[tex]\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} [/tex]
