Study the flowing pattern:  $$\dfrac{1}{1 \times 2}= \dfrac{1}{2}$$  $$\dfrac{1}{1 \times 2}+ \dfrac{1}{2 \times 3}= \dfrac{2}{3}$$  $$\dfrac{1}{1 \times 2}+ \dfrac{1}{2 \times 3}+ \dfrac{1}{3 \times 4}= \dfrac{3}{4}$$  $$\dfrac{1}{1 \times 2}+ \dfrac{1}{2 \times 3}+ \dfrac{1}{3 \times 4}+ \dfrac{1}{4 \times 5}= \dfrac{4}{5}$$  $$\dfrac{1}{1 \times 2}+ \dfrac{1}{2 \times 3}+ \dfrac{1}{3 \times 4}+ \dfrac{1}{4 \times 5}+ \dfrac{1}{5 \times 6}= \dfrac{5}{6}$$  Given that $$\dfrac{1}{1 \times 2}+ \dfrac{1}{2 \times 3}+ \dfrac{1}{3 \times 4}+ \cdots + \dfrac{1}{2011 \times 2012}= \dfrac{k}{k+2}$$ where $$k$$ is a  positive number, find the value of $$k$$ 
Overseas Competition->Knowledge Point->Calculation Modules->Splitting Terms and Finding the General Term->Splitting Terms of Fractions