Powers and Roots
of the
Simple Geometric Series
Consider the most basic geometric series
(Eq'n 1)
for x<1. We require that last criterion to ensure that the sum converges on some finite value.
We now ask what we get when we multiply
that series by itself. What is S2? To answer that question we apply
the process of multiplication by way of partial products, displaying the partial
products in a matrix:
When we sum up those partial products,
proceeding diagonally from upper right to lower left, we get
(Eq'n 2)
We could also have obtained that result
if we had simply differentiated the series, term by term, with respect to x, as
we can see if we differentiate Equation 1:
(Eq'n 3)
If we multiply Equation 2 by Equation 1,
we will obtain the cube of S. In that case our partial product matrix looks like
this:
When we sum up those partial products,
we get
(Eq'n 4)
Again, we could have obtained that
result more easily by differentiating Equation 1:
(Eq'n 5)
An examination of the partial product
matrices above and a contemplation of how we add their elements should suffice
to convince us that the coefficients of the m-th power of S comprise the set of
the (m-2)-nd order Gaussian sums; that is, we have in general by induction
(Eq'n 6)
Because the powers of S themselves
comprise series, we expect that the roots of S will also comprise series. We
can't find the roots of S by direct calculation, as we did with the powers, so
we proceed by assuming that we can express a given root of S as a Taylor series,
(Eq'n 7)
in which we calculate the n-th
coefficient by
(Eq'n 8)
evaluated at x=0.
In devising the series for the square
root we have
(Eq'n 9)
which we differentiate repeatedly. We
thus obtain the coefficients:
(Eq'n 10)
If we apply that process to subsequent
roots of S, we get
(Eq'n 11)
In general, then, for the r-th root of S
we have for the n-th coefficient of the Taylor series representing the root
(Eq'n 12)
So now we have the means to calculate
the integer powers and roots of any geometric series.
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