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The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
As there is zero X n+1 or X −1 in (1 + X) n, one might extend the definition beyond the above boundaries to include () = when either k > n or k < 0. This recursive formula then allows the construction of Pascal's triangle , surrounded by white spaces where the zeros, or the trivial coefficients, would be.
In other words, K[X] has the following universal property: For every ring R containing K, and every element a of R, there is a unique algebra homomorphism from K[X] to R that fixes K, and maps X to a. As for all universal properties, this defines the pair (K[X], X) up to a unique isomorphism, and can therefore be taken as a definition of K[X].
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The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: [56] [57] Denote the triangle's vertices in the complex plane as a = x A + y A i, b = x B + y B i, and c = x C + y C i. Write the cubic equation () =, take its derivative, and equate the (quadratic) derivative to zero. Marden's theorem says that ...
A user will input a number and the Calculator will use an algorithm to search for and calculate closed-form expressions or suitable functions that have roots near this number. Hence, the calculator is of great importance for those working in numerical areas of experimental mathematics. The ISC contains 54 million mathematical constants.
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