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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 ...
7.2 Sum of reciprocal of factorials. 7.3 Trigonometry and ... Reciprocal of tetrahedral numbers
The number e, defined as the sum of reciprocals of factorials, is not a member of any p-adic field; but for . For p = 2 one must take at least the fourth power. [ 13 ] ( Thus a number with similar properties as e — namely a p -th root of e p — is a member of Q p {\displaystyle \mathbb {Q} _{p}} for all p .)
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 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] = + + + + = =.
The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. The number of digits in the exponential factorial of 6 is approximately 5 × 10 183 230. The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number:
The sum of the reciprocals of the product of the first n integers (the Factorials) is e, so I wonder, is the sum of the reciprocals of the product of the first n Factorials (the Superfactorials), which is 1.5868056, or the sum of the reciprocals of the sum of the first n Factorials which is 1.47608642, expressible in terms of e?
Sum of reciprocals [ edit ] I think it should be said on page that sum of reciprocals is transcendental number (as said on MathWorld) and Liouville number — Preceding unsigned comment added by 79.184.102.31 ( talk ) 17:09, 6 February 2012 (UTC) [ reply ]