\(\frac{2017}{1+2}+\frac{2017}{1+2+3}+\frac{2017}{1+2+3+4}+...+\frac{2017}{1+2+3+4+...+2016}\)
\(=2017\times\left(\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+4+...+2016}\right)\)
\(=2017\times\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{1008.2017}\right)\)
\(=2017\times2\times\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{2016.2017}\right)\)
\(=4034\times\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{2016.2017}\right)\)
\(=4034\times\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2016}-\frac{1}{2017}\right)\)
\(=4034\times\left(\frac{1}{2}-\frac{1}{2017}\right)\)
\(=4034\times\frac{2015}{4034}\)
\(=2015\)
