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Let be a natural number. For a base >, we define the sum of the factorials of the digits [5] [6] of , :, to be the following: = =!. where = ⌊ ⌋ + is the number of digits in the number in base , ! is the factorial of and
7.2 Sum of reciprocal of factorials. ... These numeric series can be found by plugging in numbers from the series listed above. Alternating harmonic series
The factorial number system is a mixed radix numeral system: the i-th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value is to be multiplied by (i − 1)!
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 ...
The Stirling numbers of the first kind sum to the factorials, and count the permutations of grouped into subsets with the same numbers of cycles. [28] Another combinatorial application is in counting derangements , permutations that do not leave any element in its original position; the number of derangements of n {\displaystyle n} items is the ...
Since ! is the product of the integers 1 through n, we obtain at least one factor of p in ! for each multiple of p in {,, …,}, of which there are ⌊ ⌋.Each multiple of contributes an additional factor of p, each multiple of contributes yet another factor of p, etc. Adding up the number of these factors gives the infinite sum for (!
This is due to the definition of the Stirling numbers of the second kind as monomials in terms of falling factorials, and the behaviour of falling factorials under the indefinite sum. Interpreting the Stirling numbers of the second kind, {+}, as the number of set partitions of [+] into parts, the identity has a direct combinatorial proof since ...
In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials of positive integers.. This is the same as their sum, with the odd-indexed factorials multiplied by −1 if n is even, and the even-indexed factorials multiplied by −1 if n is odd, resulting in an alternation of signs of the summands (or alternation of addition and subtraction ...
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