Numerical calculations indicate that the following conjecture is true $$ \sum _{n=0}^{\infty} \frac{\prod _{k=0}^{n-1} \left(x+e^{-k}\right)}{\prod _{k=0}^n \left(1+te^{-k}\right)}t^n =\frac{1}{1-xt}.\tag{1} $$ Here the $n=0th$ term of the sum is $\frac{1}{1+t}$ to avoid confucion.
Question. Is the conjecture (1) correct?
This sum emerged in certain birth and death process calculations. I have no idea how to prove it.