## Finite Summation of Integer Powers (Part 2)

(Discrete Mathematics Techniques II)

Abstract
We solve the general case of the finite-summation-of-integer-powers problem $S_p(N) = \sum_{k=1}^{N} k^p$ for arbitrary $p$, and obtain a $p$-th order recurrence relation that can be used to iteratively obtain the closed form polynomial for $S_p(N)$ for any given $p$. Source code is given for computing these polynomials using Maxima, an open-source (free) symbolic computation platform. (Note: This article generalizes the recurrence relation approach that is motivated and illustrated for small $p$ in Part 1. A direct matrix method for computing the closed form solutions is given in Part 3.)

## Finite Summation of Integer Powers (Part 1)

(Discrete Mathematics Techniques I)

Abstract
We motivate an approach that uses recurrence relations to find closed form solutions to the finite-summation-of-integer-powers problem $S_p(N) = \sum_{k=1}^{N} k^p$ for any individual $p$. The approach is illustrated for small $p$: $k, k^2, k^3, k^4$. Maxima, an open-source (free) software package, is used to demonstrate how a symbolic computation platform can speed up the accurate derivation of messy algebraic expressions.

A recurrence solution to the general case (arbitrary $p$) is developed in Part 2 along with Maxima source code. A direct (non-iterative) matrix method for solving the general case is given in Part 3 along with Maxima and Octave/Matlab source code.

## Good mathematical technique and the case for mathematical insight

Good mathematical technique can bring the solution to certain mathematical questions within reach. By a proper formulation (one that is both tractable and that generalizes readily) and the use of mechanical techniques, one can often pass from a single insight to the solution of a family of problems, and in some cases, to the solution of the general question itself. … Good mathematical technique has built within it the mathematical insight of the best of previous generations.