## 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.

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