Is it true to use as a general rule of thumb that the Gamma function always "kills" power function in a series? I mean: $$\sum_{n=1}^{\infty} \frac{C^n}{\Gamma(n)^p}<\infty$$ no matter the constant $C>0$ and power $p>0$ in the gamma function as long as they are fixed and independent of $n$. Am I correct?
Now I wonder what happens with $$\sum_{n=1}^{\infty} C^n \frac{\Gamma( a_n )^p}{\Gamma(b_n)^q}\, ?$$
How can we decide when the series is convergent? For instance if $a_n=b_n$ and $q>p>0$ I guess the series converges right? Is there a general rule/criterion one can use here? Actually, I have $a_n= an$ and $b_n=bn+c$ with $a,b,c>0$. I mean $$\sum_{n=1}^{\infty} C^n \frac{\Gamma( an )^p}{\Gamma(bn+c)^q}\, ?$$ What would be the conditions on $p,q,a,b,c$ so that the sum converges?
Thanks a lot! :D
First, simplify slightly (assuming a constant $p>0$ and $C>0$):
$$ \sum_{n=1}^{\infty} \frac{C^{n}}{\Gamma(n)} = C\sum_{n=0}^{\infty} \frac{C^{n}}{n!}.$$
Here we used the fact that $\Gamma(n)=(n-1)!$ and did a change of index.
Then, apply the ratio test.
\begin{align*} \lim_{n\to\infty} \left|\frac{\frac{C^{n+1}}{(n+1)!^{p}}}{\frac{C^{n}}{n!^p}}\right| &= \lim_{n\to\infty} \left|\frac{C^{n+1} n!^{p}}{C^{n}(n+1)!^{p}}\right| \\ &= |C| \lim_{n\to\infty} \left|\frac{n!}{(n+1)!}\right|^{p} \\ &= 0. \end{align*} So the series converges.