Challenge Level

Prove that

$$k \times k! = (k+1)! - k!$$ and sum the series

$$1 \times 1! + 2 \times 2! + 3 \times 3! +...+n \times n!$$

Did you know ... ?

A telecoping series is a 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 Series.

A telecoping series is a 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 Series.