Prove the identity $1(1!)+2(2!)+3(3!)+...+n(n!)=(n+1)!-1$ without using induction

Asked Jan 20 '20 at 22:49

Active Jan 24 '20 at 19:39

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I was asked to prove this identity using two methods. I have already completed a proof by induction, but I don't know another method to prove this and the hint I was provided doesn't make any sense to me.

The hint provided was: For the other method, try to express $k(k!)$ as the difference of factorials.

edited Jan 24 '20 at 19:39

Bill Dubuque

272,048

asked Jan 20 '20 at 22:49

ahkaashi

2

Have you heard of "telescoping"? – Daniel Fischer Jan 20 '20 at 22:50
Thanks @Bill. Quick find. – Daniel Fischer Jan 20 '20 at 22:56

Prove the identity $1(1!)+2(2!)+3(3!)+...+n(n!)=(n+1)!-1$ without using induction

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