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In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
The total amount to be subtracted is 4 + 8 + 12 + 16 + ... equal to the Abel sum of its defining series, [13] ... 3 − 4 + ⋯ = 1 / 4 as an Abel sum, and 1 ...
by the functional equation and Euler's identity. For example, e iπ = e 3iπ = −1, so both iπ and 3iπ are possible values for the complex logarithm of −1. In general, given any non-zero complex number w, any number z solving the equation = is called a complex logarithm of w, denoted .
Because d 0 − 4 d 2 + 16 d 4 = 7 and because—by the nature of the quater-imaginary system—the coefficients can only be 0, 1, 2 or 3 the value of the coefficients can be found. A possible configuration could be: d 0 = 3, d 2 = 3 and d 4 = 1. This configuration gives the resulting digit string for 7 10.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
The aliquot sum function can be used to characterize several notable classes of numbers: 1 is the only number whose aliquot sum is 0. A number is prime if and only if its aliquot sum is 1. [1] The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively. [1]
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or equivalently the Dirichlet convolution of an arithmetic function () with one:
Then the triangle is in Euclidean space if the sum of the reciprocals of p, q, and r equals 1, spherical space if that sum is greater than 1, and hyperbolic space if the sum is less than 1. A harmonic divisor number is a positive integer whose divisors have a harmonic mean that is an integer. The first five of these are 1, 6, 28, 140, and 270.