## Yet Another of Euler's FormulasIn 1737 L.Euler published the following formula On the left, we sum up the reciprocals of all integers n starting with 1 raised to the real power of s. Such a sum is known in Calculus as an infinite The explanation of the formula starts with the expression for the geometric series (which is an infinite series in its own right, convergent for |q|<1): For s>1 and integer n, 0<1/n With this we may rewrite Euler's formula in the expanded form which (convergence questions aside) is almost a reformulation of the fact that every integer can be represented as a product of prime numbers. Picking on the right from between a pair of parentheses a single term and multiplying all of them we'd get a reciprocal of an infinite product: ^{m2}3^{m3}5^{m5}...)^{s}
Unless only a finite number of terms is 1, such a reciprocal is 0 as a product of an infinite number of terms less than, say, 1/2. With only a finite number of terms different from 1 the result is a reciprocal of an integer (raised to the ## References- C.Clawson,
*Mathematical Mysteries*, Plenum Press, 1996
|Contact| |Front page| |Contents| |Up| Copyright © 1996-2018 Alexander Bogomolny 71868127 |