# Thread: [SOLVED] Factorial series question

1. ## [SOLVED] Factorial series question

factorial series.pdf

My question is, what happened to the k! in the top when the original series is simplified? How does that get canceled?

2. Originally Posted by mollymcf2009

factorial series.pdf

My question is, what happened to the k! in the top when the original series is simplified? How does that get canceled?
$\displaystyle \frac{k!}{(k+3)!} = \frac{k!}{(k+3)(k+2)(k+1) \cdot k!}$

so ... what happened to the $\displaystyle k!$ ?

3. Alright, so, two things,

k! = (k)(k-1)(k-2)(k-3)(k-4)(k-5)...etc

(k+3)! = (k+3)(k+3-1)(k+3-2)(k+3-3)(k+3-4)(k+3-5)

now you'll notice that, simplifying, (k+3-3) = k, and (k+3-4) = (k-1), and (k+3-5) = (k-2), etc.

so in effect, (k+3)! has all of the terms as k!, with the except of the first three terms of (k+3)!, so you can divide them out

hopefully that explains it well enough. if not lemme know