By
Assad Ebrahim, on January 19th, 2018 (12,274 views) |
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 
. Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the 
independent monomials 
(
), where the 
are unknown coefficients. Using the recurrence relation 
, we obtain a linear combination of the monomials, which reduces to an easily solvable 
-by- triangular linear system in the unknown coefficients 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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