I would like to understand how can I write down the expression for the following series: $$S_0=\sum_{k=2}^{\infty}(-1)^kA^{k}\frac{\Gamma(k-3/2)}{\Gamma(k+1)}.$$ I have seen related topics on this site but I still do not understandt how should I start to find the sum. It is worth mentioning that $A>0$ is the parameter and the series is assumed to be convergent.
2026-03-27 18:19:14.1774635554
Series with gamma functions
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Hint: One may write the expansion of $(1+y)^\alpha$ by Binomial as $$(1+y)^\alpha=\sum_{n=0}^\infty\dfrac{\Gamma(n-\alpha)}{\Gamma(-\alpha)}\dfrac{(-y)^n}{n!}$$ here use $\alpha=\dfrac32$ and $y=A$.