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Apr 5, 2020 at 9:44 comment added PiE Yes. You had a great answer. Thank you.
Apr 5, 2020 at 8:59 comment added Eric Towers @PiE : Twice you write the sum is $1$ in your question, unconditionally on $k$. I've also more thoroughly answered your question and explained why you were having numerical precision trouble.
Apr 5, 2020 at 8:55 comment added PiE The series is the ss probabilities of an M/D/1 queue so they should sum to 1 for $k \rightarrow \infty$.
Apr 5, 2020 at 3:03 history answered Eric Towers CC BY-SA 4.0