SUMMARY
The series $ \sum_{s}^{} \frac{(2s-1)!}{(2s)!(2s+1)}$ is convergent, as established through the application of Stirling's asymptotic formula for both regular and double factorials. The relevant formula is $n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n$, which simplifies the factorial expressions involved. Additionally, a less known Stirling approximation for double factorials can be utilized: $$n! \approx \Bigl(\frac2\pi\Bigr)^{\frac14(1-\cos(n\pi))}\sqrt\pi n^{(n+1)/2}e^{-n/2}$$. This approach, along with identities from Wolfram MathWorld, provides a clear path to demonstrating the convergence of the series.
PREREQUISITES
- Understanding of Stirling's asymptotic formula for factorials
- Familiarity with double factorial notation and properties
- Knowledge of series convergence tests, including the ratio test
- Basic calculus skills, including differentiation and integration
NEXT STEPS
- Study the derivation and applications of Stirling's asymptotic formula
- Learn about the properties and applications of double factorials
- Explore convergence tests for series, focusing on the ratio and integral tests
- Review advanced integration techniques as outlined in the provided PDF resource
USEFUL FOR
Mathematicians, students studying advanced calculus or analysis, and anyone interested in series convergence and factorial approximations.