Closed-Form Expessions for Summations

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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:


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 .


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 :


We can use binomial coefficients to represent the difference in powers in terms of a summation over the polynomial components:

Solving for :