I am having some difficulties to find the solution for this sum of fractions with factorials in denominator. Any ideas how to solve this?
$$\sum_{i=1}^{2016}\frac{1}{i!+(i+1)!+(i+2)!} =\frac{1}{1!+2!+3!}+\frac{1}{2!+3!+4!}+...+\frac{1}{2016!+2017!+2018!}=?$$
I have used the following formula as a one way of solving it, but didn't get good results. $$\frac{1}{n!+(n+1)!+(n+2)!}=\frac{1}{n!(n+2)^2}$$
It turns out that we can solve/represent the solution using hypergeometric functions. Can anyone suggest the elementary way of solving/simplifying it?