Thank you for your replies. Actually DiscretePlot with WorkingPrecision->50 did worked. To show the value, I used the N operator suggested by Gianluca. Thank you!
The imprecision may be due to floating-point arithmetic. You can switch to arbitrary precision arithmetic by giving N with an explicit number of digits for the output:
DiscretePlot[N[s[10/9, n], 16], {n, 40, 80}]
I did not try, because you did submit any copy-paste-able code:
but try adding
WorkingPrecision -> 50
at the end of your plot command.
