Sum of Integer Powers (Part 1)

By , on January 21st, 2018 (52,202 views) | Leave your comment!
Topic: Maths--Technical

2nd ed. January 21, 2018; 1st ed. Feb 8th, 2010

Abstract
This three part paper explores solving the sum of powers problem S_r(n) = \sum_{k=1}^{n} k^r using discrete maths techniques (recurrence relations, matrix systems) to obtain a solution polynomials whose coefficients turn out to be exactly the Bernoulli numbers B_n.
Part 1 (this paper) solves the problem using recurrence relations in a way which a high school student could emulate for small r. In Part 2, we develop a general recursive solution that works for arbitrary r, from which we can build a table of values to assist in finding the coefficients of the solution polynomial, coefficients that are precisely the Bernoulli numbers discovered in 1713. In Part 3, we show how by transforming the problem into a linear system, we may obtain a direct (non-recursive) solution which directly calculates the Bernoulli number for any power r. Source code is provided for all solutions.

Readers who are interested in this topic are referred also to lovely paper by Bearden (March 1996, American Mathematical Monthly), which tells the mathematical story and fills in the history (thanks to a reader for this great reference).
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Sum of Integer Powers (Part 2)

By , on January 20th, 2018 (21,796 views) | 4 comments - Leave a comment!
Topic: Maths--Technical

(Discrete Mathematics Techniques II)

1st ed. Feb 8th, 2010

Abstract
We continue the 3-part paper exploring how one might solve for themselves the general case of the sum-of-integer-powers problem S_p(N) = \sum_{k=1}^{N} k^p for arbitrary p, the coefficients of whose solution are the famous Bernoulli numbers (1716). In this paper we show to how 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. Continue reading this article…

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Sum of Integer Powers (Part 3)

By , on January 19th, 2018 (12,751 views) | 2 comments - Leave a comment!
Topic: Mathematics, Maths--Technical

(Discrete Mathematics Techniques III)

1st ed. Apr 2nd, 2010

Abstract
This is the last in the 3-part series of articles on finding for oneself the solution to the sum of integer power problem, and in the process discovering the Bernoulli numbers. In Part 3 (this paper), 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.

A lovely paper by Bearden (March 1996, American Mathematical Monthly), which was shared with me by a reader, tells the mathematical story nicely, with much of the history filled in.

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The Place of Insight, Technique, and Computing in Mathematics

By , on January 31st, 2010 (8,488 views) | Leave your comment!
Topic: Education, Maths--Philosophy

The mathematician Alfred North Whitehead1 observed that “[Advancement occurs] by extending the number of important operations which we can perform without thinking of them.” (Introduction to Mathematics, 1911 2) This is certainly true in mathematics where the development of judicious notation, accompanied by good mathematical technique, extends the capability to perform chains of complex reasoning accurately and efficiently. Through proper problem formulation (tractable yet generalizable), one can sometimes pass from a single insight to the solution of a family of problems, and in some cases, to the solution to the general question itself.3

Here, mathematical computing can provide a useful benefit: helping to efficiently explore conjectures, dispatch with false directions, and save time during the development, error-checking and validation stages of obtaining general results. In industry, where specific or semi-general results are needed fast, such tools allow rapidly working up the required material and providing the necessary certainty before the fully general results or complete proof are ready.

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  1. Whitehead was the major collaborator with Bertrand Russell in the strenuous 10 year attempt, ultimately unsuccessful, at driving through the logicist program in Mathematics, i.e. reducing the entire body of mathematics to a fixed system of logic. The program of logicial reductivism, of which this was perhaps the last major attempt, and certainly one of the best known and most influential, was put to rest by Godel’s discovery of the essential incompleteness of every sufficiently strong logical system (proved in his Incompleteness Theorem). In this, he establishes that any logical system sufficiently strong to obtain arithmetic will be able to generate statements that the system cannot prove. ↩
  2. Whitehead claimed in the original that it is Civilization that advances in this way. I have reduced the claim for the purpose of this article. ↩
  3. Fields Medalist Terence Tao has written a short piece that describes the role of rigor and the value of mathematical technique in the training of a mathematician. In the online discussion of this article, he adds two particularly interesting remarks: the first concerns the difference between the training pathways of physicists and engineers versus mathematicians that acknowledges that the final destination is the same, but the training route is different (pre-rigorous, post-rigrous). He then speculates on the observation that the two pathways are not the same, and that the order in which one traverses them influences the final outcome, and he makes the analogy with the order of learning languages. ↩
(8,488 views) | Leave your comment!

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