<s>[INST] The values of the parameters for which $\frac{ab}{c}+\frac{a(a+1)b(b+1)}{2!c(c+1)}+\frac{a(a+1)(a+2)b(b+1)(b+2)}{3!c(c+1)(c+2)}+\dots$ converges. Determine the values of the parameters for which $\frac{ab}{c}+\frac{a(a+1)b(b+1)}{2!c(c+1)}+\frac{a(a+1)(a+2)b(b+1)(b+2)}{3!c(c+1)(c+2)}+\dots$ converges.
Then $\frac{a_{n+1}}{a_n}=\frac{(a+n)(b+n)}{(n+1)(c+n)}$, I guess then I should use Raabe's test, but I don't know how to write $\frac{(a+n)(b+n)}{(n+1)(c+n)}$ in form of $1-\frac{p}{n}$. Any help? Thanks.
 [/INST] $$
\frac{(a+n)(b+n)}{(c+n)(1+n)} = 1 - \frac{1+c-a-b}{n} + O(1/n^2)
$$
as $n \to \infty$.
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