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Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

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Weekly Challenge 36: Seriesly

Stage: 5 Short Challenge Level: Challenge Level:1


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.