Closed-Form Expessions for Summations
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Abstract
Some mathematical concepts are most intuitively described using summations.
To efficiently calculate and work with the results of such summations, it
may be necessary to represent them as a closed-form expression.
The following describes a method for finding closed-form expressions for
summations of polynomials 
of order 
:
Method
The first thing to note is that a summation over a polynomial may be decomposed into a summation over the sum of polynomial components:
To discover a closed-form expression for 
we represent
as a summation. This summation must consist of
with 
to not exceed the value of .
By iteratively calculating and subtracting for 
,
we are left with a closed-form expression for .
Example
Let's have a look at how a closed-form expression for 
may be derived
without prior knowledge of for 
.
We begin by representing 
as a summation. We can represent any power
by "integrating" over the differences in consecutive values:
For 
:
For 
:
For 
:
Generalization
We can use binomial coefficients to represent the difference in powers 
in terms
of a summation over the polynomial components:
Solving for :
