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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.
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 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 interaction of two factors with s 1 and s 2 levels, respectively, has (s 1 −1)(s 2 −1) degrees of freedom. The formula for more than two factors follows this pattern. In the 2 × 3 example above, the degrees of freedom for the two main effects and the interaction — the number of columns for each — are 1, 2 and 2, respectively.
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are O, A and B.
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In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...