Convergence of the surface charge density Fourier series expansion

[thesaruman]

Homework Statement

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

Homework Equations

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

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
$$u_n = \frac{4 s+3}{\sqrt{\pi } \sqrt{s}}.$$
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...

 

[Dick]

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.