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$\begingroup$ The series is the ss probabilities of an M/D/1 queue so they should sum to 1 for $k \rightarrow \infty$. $\endgroup$
– PiE
Commented Apr 5, 2020 at 8:55
1

$\begingroup$ @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. $\endgroup$
– Eric Towers
Commented Apr 5, 2020 at 8:59
$\begingroup$ Yes. You had a great answer. Thank you. $\endgroup$
– PiE
Commented Apr 5, 2020 at 9:44

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