How to test the series for convergence involving repeating sequences in the denominator and numerator?

Question

I have to test the following factorial for convergence:
$$sum_{n=0}^infty frac{7cdot15cdot23cdotcdotcdotcdotcdot(8n-1)}{8cdot17cdot26cdotcdotcdotcdotcdot(9n-1)}$$
What I have attempted to do is use the ratio test as such:
$$lim_{nto 0}lvert (frac{8n+7}{9n+8})(frac{9n-1}{8n-1})rvert = 1$$
This means that the use of this test is inconclusive. I also feel as though the steps I made in this is incorrect. Something about my work does not seem right, I cannot piece it together.

Claude Leibovici · Answer

You have
$$a_n =frac {prod_{i=1}^n(8i-1) } {prod_{i=1}^n(9i-1) }=left(frac{8}{9}right)^n,frac{Gamma left(frac{8}{9}right)}{Gamma left(frac{7}{8}right)},frac{Gamma left(n+frac{7}{8}right)}{Gamma left(n+frac{8}{9}right)}$$
$$frac {a_{n+1}}{a_n}=frac{8 n+7}{9 n+8} quad to frac{8}{9}$$

Robert Israel · Answer

Your ratio is wrong.  Note that $9n-1$ and $8n-1$ are factors in $a_n$ as well as $a_{n+1}$, so they should cancel out when you take $a_{n+1}/a_n$.

How to test the series for convergence involving repeating sequences in the denominator and numerator?

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