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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 ...
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.
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.
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.
For the avoidance of ambiguity, zero will always be a valid possible constituent of "sums of two squares", so for example every square of an integer is trivially expressible as the sum of two squares by setting one of them to be zero. 1. The product of two numbers, each of which is a sum of two squares, is itself a sum of two squares.
Near-doubles: Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact 6 + 6 = 12 by adding one more, or from 7 + 7 = 14 but subtracting one. [36] Five and ten: Sums of the form 5 + x and 10 + x are usually memorized early and can be used for deriving other facts. For example, 6 + 7 = 13 can be derived from 5 + 7 = 12 by adding one ...
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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.