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: Convergence of the surface charge density Fourier series expansion

  1. Dec 12, 2008 #1
    1. The problem statement, all variables and given/known data

    Test the convergence of the series for the surface charge density:
    [tex]\sum^{\infty}_{s=0}(-1)^s(4s+3)\frac{(2 s -1)!!}{(2s)!!}[/tex]

    2. Relevant equations

    [tex](2s-1)!! = \frac{(2s)!}{2^s s!};[/tex]
    [tex](2s)!! = 2^s s![/tex]
    Stirling's asymptotic formula for the factorials:
    [tex]s! = \sqrt{2 \pi s}s^s \exp{(-s)}.[/tex]

    3. The attempt at a solution

    Well, I couldn't show that this is a monotonically decreasing series. So I thought I could demonstrate that this is an absolutely convergent series. First, I used the ratio and root tests, but they were inconclusive. Then I used the Stirling asymptotic formula with the double factorials converted to simple ones, and obtained
    [tex]u_n = \frac{4 s+3}{\sqrt{\pi } \sqrt{s}}.[/tex]
    I used again the ration and root tests, but again they showed inconclusive results. Finally, I used the MacLaurin test showed that this series is not absolutely convergent.
    I think that this is not the case, because this is the result of a Fourier series expansion of the surface charge density of the electrostatic two hemisphere problem.
    Does anyone has any idea, please...
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Dec 12, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    I get the same thing you do as an approximation for u_n. That's not only increasing, it's unbounded. The series can't converge.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook