What would the closed form expression of the summation of $k \times k!$ from $k=0$ to $n$ be?

Question

$$\sum_{k=0}^n k\ k! = {} ?$$ My teacher assigned this for homework, and I’ve been trying to figure this out, but I couldn’t think of any convenient generating functions or binomial terms or anything. Any hints or guidance would be extremely appreciated.

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score 4 · Answer 1 · answered Oct 12 '22 at 03:29

4

Well, $k\cdot k!=(k+1-1)\cdot k!=(k+1)!-k!$, so it is just a telescoping sum. The sum will be $(n+1)!-1$.

answered Oct 12 '22 at 03:29

Potitov06

340

score 4 · Answer 2 · edited Oct 12 '22 at 03:35

4

Hint:

$k \cdot k!=(k+1-1) \cdot k!=(k+1)!-k!$

Now do telescopic summation.

edited Oct 12 '22 at 03:35

Rócherz

3,976

answered Oct 12 '22 at 03:30

Z Ahmed

43,235

What would the closed form expression of the summation of $k \times k!$ from $k=0$ to $n$ be?

2 Answers2