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 of summations:
Method
The first thing to note is that a summation over a polynomial may be decomposed into a sum over the summations of the individual polynomial components:
To discover a closed-form expression for 
we will represent
as a summation, which must consist of 
with 
. 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 :
