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A formula that permits to express one class of finite sums in explicit form.

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Deduce the recurrent formula for calculation of the finite sums of natural numbers in a natural power:

1^p + 2^p + 3^p + ... + N^p

https://brainmass.com/math/calculus-and-analysis/formula-permits-express-one-class-finite-sums-explicit-form-8726

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Solution.

We denote = p SN , as well as all similar sums with 1 k  p - 1 as k SN .
We plot then the graph of the function y = x p and mark integer-valued points to build a set of rectangles with the heights equal to n p and the unit length bases.

It's clear that the total sum of the rectangles ...

Solution Summary

A recurrent formula is deduced. The solution is detailed and well presented. A graph is included.

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