This is a typical telescoping sequence. First, we note that: $$\frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$$

Therefore, we have$$\begin{align*} & \frac{1}{1\times 2} + \frac{1}{2\times 3} + \cdots + \frac{1}{2017\times 2018} \\ = & \Big(\frac{1}{1} - \frac{1}{2}\Big)+ \Big(\frac{1}{2} - \frac{1}{3}\Big)+\cdots + \Big(\frac{1}{2017} - \frac{1}{2018}\Big)\\ =& \frac{1}{1} - \frac{1}{2018}\\ =&\boxed{\frac{2017}{2018}} \end{align*}$$