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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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.

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