# Simplifying Factorials: General Guidelines for Calculus III

In summary, according to the author, if you want to use the ratio test to check if a series converges or diverges, you need to include parentheses to cancel out the ! symbol.
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.

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?

matt grime said:
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)

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?

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.

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?

(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.

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.

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

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

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

## 1. What is a factorial?

A factorial is a mathematical operation that represents the product of all positive integers less than or equal to a given number. For example, 5 factorial (written as 5!) is equal to 5 x 4 x 3 x 2 x 1 = 120.

## 2. How do I write factorials using mathematical notation?

Factorials are written using the exclamation mark (!) after a number. For example, 5! represents 5 factorial.

## 3. How do I simplify factorials?

To simplify a factorial, you can use the following formula: n! = n x (n-1)!. This means that you can break down a factorial into a smaller factorial until you reach a factorial that you can easily calculate.

## 4. Can factorials have decimal or negative numbers?

No, factorials are only defined for positive integers. Decimal or negative numbers do not have factorial values.

## 5. What is the largest factorial that can be calculated?

The largest factorial that can be calculated depends on the computing power available. However, most calculators can only handle factorials up to 69! due to the limit of 16-digit accuracy.

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