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# expandGeomSeries -- Expand a geometric series to a specified degree.

- Usage:
    `pSeries = expandGeomSeries(f,n)`
- Inputs:
  - f, a ring element, Ratio of the geometric series to be expanded.
  - n, an integer, Degree which to expand the geometric series.
- Outputs:
  - pSeries, a ring element, Power series representation of the geometric series.

## Description

If the user supplies a list instead of a RingElement as the first argument, the return value is the product of all the each of the geometric series expanded to degree n obtained by calling expandGeomSeries on each element of the list.

    i1 : A = ZZ[S,T_0,T_1]

    o1 = A

    o1 : PolynomialRing

    i2 : f = expandGeomSeries(S^2*T_0^8,10)

          10 40    8 32    6 24    4 16    2 8
    o2 = S  T   + S T   + S T   + S T   + S T  + 1
             0       0       0       0       0

    o2 : A

    i3 : g = expandGeomSeries(S^4*T_1^15,10)

          8 30    4 15
    o3 = S T   + S T   + 1
            1       1

    o3 : A

    i4 : h = expandGeomSeries({S^2*T_0^8,S^4*T_1^15},10)

          10 40    10 24 15    10 8 30    8 32    8 16 15    8 30    6 24
    o4 = S  T   + S  T  T   + S  T T   + S T   + S T  T   + S T   + S T   +
             0        0  1        0 1       0       0  1       1       0
         ------------------------------------------------------------------------
          6 8 15    4 16    4 15    2 8
         S T T   + S T   + S T   + S T  + 1
            0 1       0       1       0

    o4 : A

    i5 : B = A/(first gens A)^11

    o5 = B

    o5 : QuotientRing

    i6 : substitute(f*g,B) == h

    o6 = true

## Ways to use expandGeomSeries:

- `expandGeomSeries(List,ZZ)`
- `expandGeomSeries(RingElement,ZZ)`

## For the programmer

The object expandGeomSeries is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/DGAlgebras.m2:2702:0.