Factorial Equivalencies Calculus

Question

Why is:

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

Using this, I can deduce that:

(n+1)! = (n)(n+1)(n-1)!

I am working with Calculus Sequences. The actual problem is:

Find whether the Sequence converges or diverges:

{(2n-1)!/(2n+1)!}

Answer 1

Note that $$ (2n+1)!=(2n+1)(2n+1-1)(2n+1-2)\cdots 1 $$ And $$ (2n-1)!=(2n+1-2)\cdots 1 $$ So $$ (2n+1)!=(2n+1)(2n)(2n-1)! $$

Answer 2

We know that $n!=1\cdot2\cdot3\ldots(n-1)\cdot n$. If you divide both sides through by $n$, the right hand side becomes the product of $1\cdot2\cdot3\ldots$ up to $(n-1)$. This is $(n-1)!$ so we can see that $n!=n\cdot(n-1)!$ for all natural numbers $n$. If you set substitute $n$ with $2n$, your equation holds.

I'm not sure about the downvote though perhaps it's cause some people define $n!$ as $n(n-1)!$, which would make your statement obvious.