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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

    Dick

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    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.
     
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