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Generalizing this argument, any infinite sum of values of a monotone decreasing positive function of (like the harmonic series) has partial sums that are within a bounded distance of the values of the corresponding integrals. Therefore, the sum converges if and only if the integral over the same range of the same function converges.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The harmonic numbers roughly approximate the natural logarithm function [2]: 143 and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers.
This was proved by Leonhard Euler in 1737, [1] and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series).
The Kempner series is the sum of the reciprocals of all positive integers not containing the digit "9" in base 10. Unlike the harmonic series, which does not exclude those numbers, this series converges, specifically to approximately 22.9207 . A palindromic number is one that remains the same when its digits are reversed.
2.1 Harmonic numbers. ... By taking to be the partial sum function associated to some sequence, this leads to the summation by parts formula. Examples. Harmonic ...
In zeta function regularization, the series = is replaced by the series =. The latter series is an example of a Dirichlet series. When the real part of s is greater than 1, the Dirichlet series converges, and its sum is the Riemann zeta function ζ(s).
= + = + +, which has a sum of the natural logarithm of 2, while the sum of the absolute values of the terms is the harmonic series, = = + + + + +, which diverges per the divergence of the harmonic series, [28] so the alternating harmonic series is conditionally convergent.