The question is to find whether the series $$\sum_{n=1}^{\infty}\frac {(p+1)(p+2)...(p+n)}{(q+1)(q+2)...(q+n)},$$ given that $q<p+1,$ is convergent (absolutely or conditionally) or divergent. I have already succeeded on proving that this is divergent through the Corollary of the Raabe's test. But I would like to ask some hints on how to use other possible methods and maybe a shorter proof since mine is quite lengthy though. (especially Limit Comparison Test) maybe on the series that I would compare it to.
2026-04-02 00:21:57.1775089317
Show that the infinite series is divergent (Alternate solutions)
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Since $q < p+1$, $$\begin{align} \frac{(p+1)(p+2)\dots(p+n)}{(q+1)(q+2)\dots (q+n)} &> \frac{(p+1)(p+2)\dots(p+n)}{(p+1+1)(p+1+2)\dots (p+1+n)} \\ &= \frac{(p+1)(p+2)\dots(p+n)}{(p+2)(p+3)\dots (p+n+1)} \\ &= \frac{p+1}{p+n+1} \operatorname*{\sim}_{n\to\infty} \frac{p}{n} \end{align}$$ so the series diverges by comparison with the Harmonic series.