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The sum of the reciprocals of the powerful numbers is close to 1.9436 . [4] The reciprocals of the factorials sum to the transcendental number e (one of two constants called "Euler's number"). The sum of the reciprocals of the square numbers (the Basel problem) is the transcendental number π 2 / 6 , or ζ(2) where ζ is the Riemann zeta ...
While the partial sums of the reciprocals of the primes eventually exceed any integer value, they never equal an integer. One proof [6] is by induction: The first partial sum is 1 / 2 , which has the form odd / even . If the n th partial sum (for n ≥ 1) has the form odd / even , then the (n + 1) st sum is
The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum, as Leonhard Euler proved in 1737.. Like rational numbers, the reciprocals of primes have repeating decimal representations.
The reciprocal Fibonacci constant ψ is the sum of the reciprocals of the Fibonacci numbers: = = = + + + + + + + +. Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.
7.2 Sum of reciprocal of factorials. ... 7.4 Reciprocal of tetrahedral numbers. 7.5 Exponential and logarithms. 8 See also. 9 Notes. 10 References. Toggle the table ...
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).
A taxicab number is the smallest integer that can be expressed as a sum of two positive third powers in n distinct ways. The Riemann zeta function is the sum of reciprocals of the positive integers each raised to the power s, where s is a complex number whose real part is greater than 1.
The harmonic number with = ⌊ ⌋ (red line) with its asymptotic limit + (blue line) where is the Euler–Mascheroni constant.. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] = + + + + = =.