Learning from Mistakes: Ratio Test Problem Solving

[mmont012]

Homework Statement

I'm reviewing for a test and working on the practice problems for the ratio test that Pauls Online Notes gives. So here is given problem:

Here is his solution for the problem:

2. The attempt at a solution
I worked this out before I looked at the solution and I got it wrong. I had it right up until the 4th step. Can someone explain this step to me? I don't understand why the (n-1)! is broken up into (n-1)(n-2)! Isn't this supposed to be the decomposition of (n-2)!? I'm sorry if this is a stupid question, I'm just trying to learn from my mistakes so that I won't do this again.

Thanks for stopping by!

 

[WWGD]

Not sure I understand your question; (n-1)! =(n-1)(n-2)...2.1= (n-1)[(n-2)(n-3)...2.1]= (n-1)(n-2)! is a decomposition of (n-1)!

 

[mmont012]

Then what is the decomposition on (n-2)! ?

 

[WWGD]

In this format, it is (n-2)(n-3)! . In general, (n-k)!= (n-k)(n-k-1)! if n>k, to avoid a -1!.

 

[mmont012]

Thank you so much! This helps me out tremendously!

 

[WWGD]

Glad it helped, mmont012.