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The symbolical representation of the results of this paper is much facilitated by the introduction of a separate symbol for the product of alternate factors, , if be odd, or if be odd [sic]. I propose to write n ! ! {\displaystyle n!!} for such products, and if a name be required for the product to call it the "alternate factorial" or the ...
The following methods apply to any expression that is a sum, or that may be transformed into a sum. Therefore, they are most often applied to polynomials , though they also may be applied when the terms of the sum are not monomials , that is, the terms of the sum are a product of variables and constants.
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
More carefully bounding the sum both above and below by an integral, using the trapezoid rule, shows that this estimate needs a correction factor proportional to . The constant of proportionality for this correction can be found from the Wallis product , which expresses π {\displaystyle \pi } as a limiting ratio of factorials and powers of two.
The numbers or the objects to be added in general addition are collectively referred to as the terms, [6] the addends [7] [8] [9] or the summands; [10] this terminology carries over to the summation of multiple terms. This is to be distinguished from factors, which are multiplied. Some authors call the first addend the augend.
The Brahmagupta–Fibonacci identity states that the product of two sums of two squares is a sum of two squares. Euler's method relies on this theorem but it can be viewed as the converse, given n = a 2 + b 2 = c 2 + d 2 {\displaystyle n=a^{2}+b^{2}=c^{2}+d^{2}} we find n {\displaystyle n} as a product of sums of two squares.
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A Cabtaxi number is the smallest positive number that can be expressed as a sum of two integer cubes in n ways, allowing the cubes to be negative or zero as well as positive. The smallest cabtaxi number after Cabtaxi(1) = 0, is Cabtaxi(2) = 91, [ 5 ] expressed as: