Convergence of the surface charge density Fourier series expansion

In summary, the conversation is about testing the convergence of a series for the surface charge density using various methods. The speaker uses the ratio and root tests, Stirling's asymptotic formula, and the MacLaurin test, but is unable to prove convergence. They mention that the series is the result of a Fourier series expansion in the electrostatic two hemisphere problem. Another person confirms that the series is unbounded and therefore cannot converge.
  • #1
thesaruman
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0

Homework Statement



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]

Homework 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]

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...
 
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  • #2
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.
 

Related to Convergence of the surface charge density Fourier series expansion

1. What is the surface charge density Fourier series expansion?

The surface charge density Fourier series expansion is a mathematical technique used to represent a periodic function, such as a surface charge density, as an infinite sum of sinusoidal functions. This expansion allows for a more efficient and accurate representation of the original function.

2. How does the convergence of the Fourier series expansion relate to surface charge density?

The convergence of the Fourier series expansion indicates how well the infinite sum of sinusoidal functions approximates the original function. In the case of surface charge density, a higher convergence means that the Fourier series expansion is a better representation of the actual charge distribution on the surface.

3. What factors affect the convergence of the surface charge density Fourier series expansion?

The convergence of the Fourier series expansion is influenced by the periodicity of the function, the smoothness of the function, and the number of terms used in the expansion. A highly periodic and smooth function with a large number of terms will have a higher convergence compared to a less periodic and rough function with a small number of terms.

4. How is the surface charge density Fourier series expansion used in practical applications?

The Fourier series expansion of surface charge density is commonly used in electromagnetics and electrostatics to analyze and design complex systems. It allows for the efficient calculation of electric fields and potentials, as well as the prediction of the behavior of electromagnetic waves and charge distributions.

5. What are the limitations of the surface charge density Fourier series expansion?

One limitation of the Fourier series expansion is that it assumes the function is periodic, which may not always be the case in real-world applications. Additionally, the convergence of the expansion can be slow for functions with sharp variations, making it difficult to accurately represent these types of functions with a finite number of terms.

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