How to cancel factorials in power series problems?

[shanepitts]

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?

Thank you in advance.

 

[Mark44]

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.

How is this done?

Thank you in advance.

 

[statdad]

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. :)

 

[Mentallic]

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