## 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.