1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Expressing double factorial for odd integers

  1. Sep 17, 2011 #1
    1. The problem statement, all variables and given/known data

    Express [itex]\frac{1}{(2n+1)!!}[/itex] as the following


    where 0 <= n <= infinity

    2. Relevant equations

    The double factorial for odd integers is

    (2n+1)!! = (2n+1)(2n-1)(2n-3)....1 where 0 <= n <= infinity

    3. The attempt at a solution

    Visited this website http://mathworld.wolfram.com/DoubleFactorial.html and scrolled
    down to the expression for

    [itex](-2n-1)!! = \frac{(-1)^{n}2^{n}n!}{2n!}[/itex]

    performing the sign inverse, i got

    [itex](2n+1)!! = \frac{(1)^{n}-2^{n}n!}{-2n!}[/itex]
  2. jcsd
  3. Sep 17, 2011 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    It's not at all clear what the x2n+1 is doing in there.

    As for (2n+1)!!, start by working with (2n+1)! and see what needs to be canceled out.

    [itex]\displaystyle (2n+1)!=(2n+1)(2n)(2n-1)(2n-2)(2n-3)(2n-4)\dots(5)(4)(3)(2)(1)[/itex]
    [itex]\displaystyle =\{(2n+1)(2n-1)(2n-3)\dots(5)(3)(1)\}\{(2n)(2n-2)\dots(4)(2)\}[/itex]

    [itex]\displaystyle =\{(2n+1)!!\}\{(2^n)(n!)\}[/itex]​

    Solve for (2n+1)!! .

    As for [itex]\displaystyle \frac{(-2)^{n}n!x^{2n+1}}{(2n+1)!}\,,[/itex] that's the same as [itex]\displaystyle \frac{2^nn!}{(2n+1)!}(-1)^{n}\,x^{2n+1}\,.[/itex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook