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A series is convergent (or converges) if and only if the sequence ... The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"; ...
The sum of the reciprocal of the primes increasing without bound. The x axis is in log scale, showing that the divergence is very slow. The red function is a lower bound that also diverges.
If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence.
Applications of the harmonic series and its partial sums include Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are needed to provide a complete range of responses, the connected components of random graphs, the block-stacking problem on how far over the edge ...
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.
The convergence to Brun's constant. In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B 2 (sequence A065421 in the OEIS).
The series of reciprocals of all prime divisors of Fermat numbers is convergent. (Křížek, Luca & Somer 2002) If n n + 1 is prime and , there exists an integer m such that n = 2 2 m. The equation n n + 1 = F (2 m +m) holds in that case. [13] [14] Let the largest prime factor of the Fermat number F n be P(F n). Then,
This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes. In many cases, the analytic function associated with a Dirichlet series has an analytic extension to a larger domain.