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

## Synopsis

• Usage:
pSeries = expandGeomSeries(f,n)
• Inputs:
• f, , Ratio of the geometric series to be expanded.
• n, an integer, Degree which to expand the geometric series.
• Outputs:
• pSeries, , 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 .