In math.stackexchange.com/questions/4464861/… I gave the first terms of the expansion. Is it inspiring ?

wolframalpha.com/input?i=Sum%5B%28%281+%2B+An%29%5E%28-1%29+‌​-+%28-1+%2B+A+%2B+An‌​%29%5E%28-1%29%29%2F‌​A%5En%2C%7Bn%2C1%2CI‌​nfinity%7D%5D

@Dirk. Whazt I suspect is that the OP started with specific values of $A$ which lead to Gaussian hypergeometric functions. Try the general case. Cheers :-)

@BrendanMcKay. No, for sure; Tehy are not continuous with respect to anything

Tryng different methods for numerical integration, no problem for $0.218 \leq x \leq 3.381$ but outside this range, all of them just explode

@GA316. You are very welcome ! You made me (feeling) younger with this problem. Do not hesitate to contact me for any clarification or whatever. I did not recat to the first version of the problem when the $n_k$ are complex since this is a very, very difficult problem. Cheers :-)

Not $2$ years but $27$ !!!