# How to manipulate factorials

I have been practicing power series problems and a lot of them include factorials. To find out if they converge or not I'll often use the ratio test. However, I never quite understood how to cancel factorials when replacing the n with n+1. i.e. the textbook has an example problem that shows that

[(n+1)!]2 ⇒ (n+1)2 (n!)

How is this done?

Mark44
Mentor
I have been practicing power series problems and a lot of them include factorials. To find out if they converge or not I'll often use the ratio test. However, I never quite understood how to cancel factorials when replacing the n with n+1. i.e. the textbook has an example problem that shows that

[(n+1)!]2 ⇒ (n+1)2 (n!)
In your example, [(n+1)!]2 means [(n+1)!] * [(n+1)!], which would be (n + 1)2(n)2(n - 1)2 ... 3222.
shanepitts said:
How is this done?

shanepitts
Homework Helper
Note that $[(n+1)!]^2 \ne \big(n+1\big)^2 \big(n!\big)$, so you will have difficulty reducing the left side to the right side. :)

shanepitts
Mentallic
Homework Helper
A slight typo, but it should be
[(n+1)!]2 ⇒ (n+1)2 (n!)2
$$(n+1)! = (n+1)\times n!$$

Hence

$$\left[ (n+1)!\right]^2 = \left[ (n+1)\times n!\right]^2$$

shanepitts
In your example, [(n+1)!]2 means [(n+1)!] * [(n+1)!], which would be (n + 1)2(n)2(n - 1)2 ... 3222.
Thanks