formal series over InfinitePolynomialRing

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The expression x[n] is an error because n is the Sage function numerical_approx rather than an integer. The notation for series that you try to use is valid when n is symbolic, i.e. var('n'), but that won't do here because x[n] will still be an error. It seems that PowerSeriesRing is for finite power series expansions with specified (possibly infinite) precision. So you can do e.g.

sage: sum(x[n]*t^n for n in range(5)) + O(t^5)
x_0 + x_1*t + x_2*t^2 + x_3*t^3 + x_4*t^4 + O(t^5)

Maybe you want a Lazy Power Series instead:

P.<x> = InfinitePolynomialRing(QQ)
L.<t> = LazyPowerSeriesRing(P)
f = L()
f.order = 1
f.aorder = 1
f.initialize_coefficient_stream(lambda ao: iter(x[n] for n in NN))

Then you can do:

sage: f.coefficients(5)
[0, x_1, x_2, x_3, x_4]
sage: f
x_0*1 + x_1*t + x_2*t^2 + x_3*t^3 + x_4*t^4 + O(x^5)
sage: f.coefficients(7)
[0, x_1, x_2, x_3, x_4, x_5, x_6]
sage: f
x_0*1 + x_1*t + x_2*t^2 + x_3*t^3 + x_4*t^4 + x_5*t^5 + x_6*t^6 + O(x^7)

Here the + O(x^7) is a bug in sage.combinat.species.series.LazyPowerSeries.__repr__ which forgets to use the name of the actual power series variable.

Actually I am not sure how one can properly manipulate such a LazyPowerSeries further. Things like multiplying by a constant already seem to be tricky.