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: Radius of convergence

  1. Mar 27, 2008 #1
    I am looking for radius of convergence of this power series:
    [tex]\sum^{\infty}_{n=1}[/tex]a[tex]_{n}[/tex]x[tex]^{n}[/tex], where a[tex]_{n}[/tex] is given below.
    a[tex]_{n}[/tex] = (n!)^2/(2n)!

    I am looking for the lim sup of |a_n| and i am having trouble simplifying it. I know the radius of convergence is suppose to be 4, so the lim sup should equal 1/4.

    here is my work (leaving lim sup as n-> inf out as i don't want to write it every line)
    I expanded out the bottom factorial:
    [(n!)(n!) / (2n)(2n-1)(2n-2)(2n-3)(2n-4)! ] ^ (1/n)
    and found that you can take out a 2 from the bottom even terms (now i don't know how to express the odd terms as a factorial):
    [(n!)(n)(n-1)(n-2)! / (2)(n)(n-1)(n-2)!(2n-1)(2n-3)(2n-5)...! ] ^ (1/n)
    and i canceled out the n! in top and bottom:
    [ (n!) / 2 (2n-1)(2n-3)(2n-5)(2n-7)..! ] ^ (1/n)
    now i am stuck..

    any help would be highly appreciated. haven't really dealt with factorials in awhile.
  2. jcsd
  3. Mar 27, 2008 #2


    User Avatar
    Science Advisor

    The way you have written that makes it very difficult to read. I presume you are using the ratio test to find the radius of convergence.

    [tex]a_{n+1}|x^{n+1}|= \frac{((n+1)!)^2}{(2(n+1))!}|x^{n+1}|[/tex]
    and you want to divide that by
    [tex]a_n|x^n|= \frac{(n!)^2}{(2n)!}|x^n|[/tex]
    Okay, just be careful to combine the corresponding parts into fractions:
    Now, you certainly should know that (n+1)!/n! = n+1 and it is not to difficult to see that (2n+2)!= (2n+2)(2n+1)(2n)! so that (2n)!/(2n+2)!= 1/((2n+2)(2n+1)). What you have reduces to
    and since 2n+2= n+1 we can cancel n+1 in numerator and denominator to get
    Now, what is the limit of that as n goes to infinity?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook