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.

Why Zero Raised to the Zero Power IS One

Updated! February 5, 2017

The value of zero raised to the zero power, $(0^0)$, has been discussed since the time of Euler in the 18th century (1700s). There are three reasonable choices: 1,0, or “indeterminate”. Despite consensus amongst mathematicians that the correct answer is one, computing platforms seem to have reached a variety of conclusions: Google, R, Octave, Ruby, and Microsoft Calculator choose 1; Hexelon Max and TI-36 calculator choose 0; and Maxima and Excel throw an error (indeterminate). In this article, I’ll explain why, for discrete mathematics, the correct answer cannot be anything other than 0^0=1, for reasons that go beyond consistency with the Binomial Theorem (Knuth’s argument).

Maxima for Symbolic Computation

Maxima is a symbolic computation platform that is free, open source, runs on Windows, Linux, and Mac, and covers a wide range of mathematical functions, including 2-D/3-D plotting and animation. Capabilities include algebraic simplification, polynomials, methods from calculus, matrix equations, differential equations, number theory, combinatorics, hypergeometric functions, tensors, gravitational physics, PDEs, nonlinear systems.  With an active developer base and responsive community, a user gets a secure future lifecycle of the product and plenty of help when dealing with problems. The result: an essential mathematical computing package for students, programmers, engineers, scientists, and mathematicians. This article will help you get started with Maxima.

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.