Copyright © University of Cambridge. All rights reserved.
'Weekly Challenge 36: Seriesly' printed from http://nrich.maths.org/
$$k \times k! = (k+1)! - k!$$ and sum the series
$$1 \times 1! + 2 \times 2! + 3 \times 3! +...+n \times n!$$
know ... ?
A telecoping series
series that can be written as the difference of two expressions in
such a way that almost all the terms cancel with the following or
preceding term leaving a few terms which can be combined to give
the sum of the series. See the Wikipedia article Telescoping