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Because it is a divergent series, it should be interpreted as a formal sum, an abstract mathematical expression combining the unit fractions, rather than as something that can be evaluated to a numeric value. There are many different proofs of the divergence of the harmonic series, surveyed in a 2006 paper by S. J. Kifowit and T. A. Stamps. [13]
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 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.
The essential thrust of a proof follows, patterned after Oresme's proof of the divergence of the harmonic series. To see the first inequality, the terms of the original series are rebracketed into runs whose lengths are powers of two, and then each run is bounded above by replacing each term by the largest term in that run. That term is always ...
Harmonic series (mathematics) divergence of the (standard) harmonic series; Highly composite number; Area of hyperbolic sector, basis of hyperbolic angle; Infinite series. convergence of the geometric series with first term 1 and ratio 1/2; Integer partition; Irrational number. irrationality of log 2 3; irrationality of the square root of 2 ...
Goldbach's original proof to Euler involved assigning a constant to the harmonic series: = =, which is divergent. Such a proof is not considered rigorous by modern standards. There is a strong resemblance between the method of sieving out powers employed in his proof and the method of factorization used to derive Euler's product formula for the ...
so that divergence is clear given the double-logarithmic divergence of the inverse prime series. (Note that Euler's original proof for inverse prime series used just the converse direction to prove the divergence of the inverse prime series based on that of the Euler product and the harmonic series.)
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.