## Finite Summation of Integer Powers (Part 3)

(Discrete Mathematics Techniques III)

Abstract
We find a direct closed-form solution, i.e. one that does not require iteration, for the general case of the finite-summation-of-integer-powers problem $S_p(N) = \sum_{k=1}^{N} k^p$. Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the $p+1$ independent monomials $a_j N^j$ ($1 \leq j \leq p+1$), where the $a_j$ are unknown coefficients. Using the recurrence relation $S_p(N+1) = S_p(N) + (N+1)^p$, we obtain a linear combination of the monomials, which reduces to an easily solvable $(p+1)$-by-$(p+1)$ triangular linear system in the unknown coefficients $a_j$ of the closed-form polynomial solution. Maxima and Octave/Matlab codes for directly computing the closed-form solutions are included in the Appendices.

## 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.)

## Finding Sums of Powers (Part 1)

Abstract
This paper uses recurrence relations to find a closed form solution to the sum-of-powers problem $S_r(N) = \sum_{k=1}^{N} k^r$ for any given integer $r$. We use Maxima, a free symbolic computation package, to crunch through messy algebraic expressions and reach a simplified closed form. A solution to the general case (arbitrary $r$) is developed in Part 2. A matrix alternative to the general case solution is given in Part 3. Source code is provided for all solutions.

## 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.

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