\(\left(1+\frac{1}{1\cdot3}\right)\left(1+\frac{1}{2\cdot4}\right)\left(1+\frac{1}{3\cdot5}\right)\cdot...\cdot\left(1+\frac{1}{2015\cdot2017}\right)\cdot\frac{1}{2016}\)
\(=\frac{4}{3}\cdot\frac{9}{8}\cdot\frac{16}{15}\cdot...\cdot\frac{4064256}{4064255}\cdot\frac{1}{2016}\)
\(=\frac{\left(2\cdot2\right)\left(3\cdot3\right)\left(4\cdot4\right)\cdot...\cdot\left(2016\cdot2016\right)}{\left(1\cdot3\right)\left(2\cdot4\right)\left(3\cdot5\right)\cdot...\cdot\left(2015\cdot2017\right)}\cdot\frac{1}{2016}\)
\(=\frac{\left(2\cdot3\cdot4\cdot...\cdot2016\right)\left(2\cdot3\cdot4\cdot...\cdot2016\right)}{\left(1\cdot2\cdot3\cdot...\cdot2015\right)\left(3\cdot4\cdot5\cdot...\cdot2017\right)}\cdot\frac{1}{2016}\)
\(=\frac{2016\cdot2}{1\cdot2017}\cdot\frac{1}{2016}\)
\(=\frac{2}{2017}\)
