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I am looking to simplify the factorial:

2(i!)/(2i)!.

I know that this could be simplified to:

2/(product of numbers between i+1 and 2i).

However I am unsure how to write that as a more simplified version of the original. To me, this seems as simple as possible.

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    What you have is hard to beat. – 2017-01-19

2 Answers 2

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You have

$$ \frac{2(i!)}{(2i)!}=\frac{2(1\cdot 2\cdot\ldots\cdot i)}{(1\cdot 2\cdot\ldots\cdot i)\cdot\bigl((i+1)(i+2)\dots(2i)\bigr)}. $$

I can't see more simplified form. Maybe $$2\prod_{k=1}^i \frac{1}{i+k}?$$ Of course, this is ecactly the same.

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If it's a mathematical notation you're looking for, then you could do this:

$$\frac{2(i!)}{(2i!)}=\frac{2(1\cdot 2\cdot\ldots\cdot i)}{(1\cdot 2\cdot\ldots\cdot i)\cdot\bigl((i+1)(i+2)\dots(2i)\bigr)}=\frac{2}{\prod_{j={i+1}}^{2i} j}=2 {(\prod_{j={i+1}}^{2i} j)}^{-1}$$

There is no other more simplified notation for the simplification, I believe.

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    Thank you for your response. I believe we are working towards a simpler equation. For example, we are asked to simplify (i+2)! / (i-1)!. Of course, this can be broken down into basic multiplication (i+2)(i+1)(i). Is there any way to write the original question as a series of products or exponents? – 2017-01-19
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    @JonathonM I'm sorry, but the term "series of products" is quite ambiguous to me. Would you like to detail it a bit? Do you mean series as in calculus? – 2017-01-19
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    I apologize. A series of products would be something like (1*2*3*4*5*6). That is a series of products from 1 to 6. Or (5*6*7*8*9*10) is a series of products from 5 to 10. Is it possible to write the original question using something similar to this, or possibly using exponents? – 2017-01-19
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    That is exaclty the meaning of $\prod_{i=1}^n \ (=1\cdot 2\cdot ... \cdot n)$. If you want to get the product of the numbers from 5 to 10, then you simply have to change the limits of $\prod$: $\prod_{i=5}^{10} \ (=1\cdot 2\cdot ... \cdot n)$. – 2017-01-19