A(x,n) = (x+1).....(x + n), which I assume is the product of all integers x+1 to x+n.  Then A(n) = n!/(x+n)!

0!*1!/(x+1)! + 1!*2!/(x+2)! + 2!*3!/(x+3)! + 3!*4!/(x+4)! + ...

The ratio of term n-1 to term n is (n-1)*n/(x+n). For any value x there is a value for n when this fraction exceeds 1, any n larger than this value also make the fraction exceed 1.  So for any specific x, the sum is divergent.

I'm not sure what to make of this as the problem seems to imply there is a closed form.