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Sum of powers of positive integers

Sums of the form: \[ \sigma_k(n)=1^k+2^k+\cdots+n^k \] are called the sum of powers of postive integers and have been studied throughout the history of mathematics. In this section we will give a historical account of these studies.

Pythagoras (570-500 BC)

Pythagoras observed that the sum of succesive powers of positive integers could be arranged as triangles. They called these sum triangular numbers.

Triangular Numbers

He noticed that two times the same triangular gives the number $n(n+1)$.

So concluded: \[ \sigma_1(n)=1+2+3+\cdots+n=\frac{n(n+1)}{2} \] He also noticed that the sum of two consecutive triangular numbers is a square number.

This can also be algebraically verified: \[ \frac{n(n+1)}{2}+\frac{(n+1)(n+2)}{2}=(n+1)^2 \]

In modern analysis courses these formulas are proofed using the induction method.

Archimedes (287 - 212 BC)

Archimedes studied the sum of powers of squares. He noticed following:

\[ 3(1^2+2^2+3^2+\cdots+n^2)=n^2(n+1)+(1+2+3+\cdots+n) \]

Consider the sum of the following serie: \[ s(n)=1+3+6+10+\cdots+\frac{n(n+1)}{2} \] Notice that geometrically this sum can be represented by a pyramid of $s(n)$ cubes.

If we add 6 of these pyramides together we have a block with $n(n+1)(n+2)$ cubes.

So we have found: \[ s(n)=1+3+6+10+\cdots+\frac{n(n+1)}{2}=\frac{n(n+1)(n+2)}{6} \]

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Copyright 2012 Jacq Krol. All rights reserved. Created June 2012; last updated August 2012.