Puzzle of the Week # 212 - Infinite Sum

The convergent sum of the following infinite sequence, in its simplest terms, is a fraction with a square number on the top and a factorial number below.


1/7 + 1/16 + 1/27 + 1/40 + 1/55 + 1/72 + … + 1/n(n+6) + …


What is it?


Footnote: the partial sum converges extremely slowly, such that if you add the first million terms you will only get the first 5 decimal places. However there exists a very nice trick to allow you to work out exactly the number the infinite sum converges upon (without having to do an infinite number of calculations!).