How to Apply Gauss's Convergence Test to a Hypergeometric Series?

[ognik]

I found the interval of convergence for a hypergeometric series as |x| < 1, now I believe that I need to apply 'Gauss's test' to check the end point(s). For $ \left| x \right|=1 $ my $ \left| \frac{{a}_{n}}{{a}_{n+1}} \right| = \left| \frac{n^2+(\gamma+1)n+\gamma}{n^2+(\alpha+\beta)n+\alpha\beta} \right|$
As far as I can see, Gauss's test says I need to get this to the form $ 1+\frac{h}{n}+\frac{{C}_{n}}{{n}^{r}}$
Dividing top and bottom by n² gets BOTH top & bottom into that form, but then I'm stuck, how do I get the values of $ \alpha,\beta,\gamma $? I can't see a way to simplify to the single level of that form?

 

[jvicens]

Hello,

Thank you for sharing your findings on the interval of convergence for a hypergeometric series. It seems like you have made good progress in your research.

To apply Gauss's test, you are correct in trying to get the ratio of consecutive terms in the series to the form of $1+\frac{h}{n}+\frac{{C}_{n}}{{n}^{r}}$. This form is known as the "general term" of a series. Once you have the general term in this form, you can use the values of h and r to determine the convergence or divergence of the series.

In order to find the values of $ \alpha,\beta,\gamma $, you can use algebraic manipulations to simplify the ratio of consecutive terms. You mentioned that dividing both the top and bottom by $n^2$ gets both the top and bottom into the desired form. This is a good start, but you can also try to factor the numerator and denominator to see if any patterns emerge. You can also use known identities or properties of the parameters $\alpha,\beta,\gamma$ to simplify the ratio further.

It may take some trial and error, but with persistence, I believe you will be able to simplify the ratio to the desired form and determine the values of $\alpha,\beta,\gamma$. I wish you the best of luck in your research.