Simplifying Factorials: General Guidelines for Calculus III

[RadiationX]

I need some general guidelines on how to simplify factorials. I'm in Calculus III

and the Prof. and unfoutunately our textbook has glossed over how to do this.

All the factorials we are dealing with now are in relation to sequences and series.

so I'm dealing with expressions that look like this:

$$\sum_{n=1}^\infty\frac{n!}{1000^nn^{1000}}$$


$$\sum_{n=1}^\infty\frac{n!}{1000^nn^{1000}}$$


If i were to use the ratio test to see if the above series converged or diverged. How would i simplify the factorials?

I know how to apply the ratio test. I need to know the general rule(s) for simplifying factorials.

If anyone knows of a link of or a free e-book or anything that would help me out i'd really appreciate it.

 

[matt grime]

What is n factorail? What is n+1 factorial? Hint: (n+1)! = (n+1) times what? Actually that's more than a hint isn't it?

 

[RadiationX]

What is n factorail? What is n+1 factorial? Hint: (n+1)! = (n+1) times what? Actually that's more than a hint isn't it?

is this correct? (n+1)!= n(n+1)

 

[RadiationX]

how would i simplify? (2(n+1))! does it equal this?(2n+2)!=(2n+1)(2n)! at what stage of the simplification process does the ! symbol go away?

 

[matt grime]

is this correct? (n+1)!= n(n+1)

no but conceivably that's a typo. what is n!, what is (n+1)! write it out for small n if need be.

 

[matt grime]

how would i simplify? (2(n+1))! does it equal this?(2n+2)!=(2n+1)(2n)! at what stage of the simplification process does the ! symbol go away?

No, and her'es the answer.

(2n+2)! = (2n+2)*(2n+1)!

who knows when it goes away since you've not said what you're trying to cancel it by.

 

[RadiationX]

I need to simplify this last week and i did not do it correctly. So my questions are stemming from using the ratio test to find if a series converges or diverges. this one for example:

$$\sum_{n=1}^\infty\frac{(2n)!}{n^n}$$ I thought that i could use the ratio test to write the following:


$$\frac{2(n+1)!}{(n+1)^{n+1}}\frac{n^n}{2n!}$$

but the above rewrite is incorrect i was told.

 

[Justin Lazear]

$$\frac{2(n+1)!}{(n+1)^{n+1}}\frac{n^n}{2n!}$$

I told you this wasn't correct because you've neglected some parentheses. I wasn't sure if you understood that you were getting at (2n+2)! and not something incorrect. As you've written it, 2(n+1)!, it equals (n+1)!*2 not the (2n+2)! = (2n+2)(2n+1)(2n)! that you want. You must write either (2n+2)! or (2(n+1))!.

--J

 

[RadiationX]

I see my mistake now. and i think that i know how to simplify this.