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How can we tell Series to treat the expansion parameter as an integer?

It is simple for Mathematica to find an asymptotic expansion for $\frac{1}{-1+p}$ as $p \rightarrow \infty$. However, if we want to restrict $p$ to be an integer and also include some terms that ...
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How to have Mathematica find asymptotics with correct asumptions

I'm tried to find a certain expansion of the Hypergeometric1F1 function using Mathematica: ...