# Factorial identity question

$\frac{2 i}{(2 i + 1)!}$ = $\frac{1}{(2 i)!}$ - $\frac{1}{(2 i + 1)!}$

Could anybody please show what it is that needs to be done on LHS to get to RHS in this identity.

coolul007
Gold Member
$\frac{2 i}{(2 i + 1)!}$ = $\frac{1}{(2 i)!}$ - $\frac{1}{(2 i + 1)!}$

Could anybody please show what it is that needs to be done on LHS to get to RHS in this identity.
(2i + 1)! = (2i+1)(2i!)

$\frac{2i}{(2i+1)!}$=$\frac{2i}{(2i+1)(2i)! }$
=$\frac{2i}{2i(2i)! + (2i)! }$
= $\frac{2i}{2i(2i)! + 2i (2i - 1)! }$
= $\frac{1}{(2i)! + (2i - 1)! }$

and now what?

$\frac{2i}{(2i+1)!}$=$\frac{2i}{(2i+1)(2i)! }$
=$\frac{2i}{2i(2i)! + (2i)! }$
= $\frac{2i}{2i(2i)! + 2i (2i - 1)! }$
= $\frac{1}{(2i)! + (2i - 1)! }$

and now what?
Nah, you've gone at that the wrong way.

Start with the question, and use coolul007's hint to get a common denominator on the right hand side.

Another way to go about: Add and subtract 1 in the numerator and club 2i and 1.

thanks acabus, all