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In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
Trachtenberg called this the 2 Finger Method. The calculations for finding the fourth digit from the example above are illustrated at right. The arrow from the nine will always point to the digit of the multiplicand directly above the digit of the answer you wish to find, with the other arrows each pointing one digit to the right.
The method of regularization using a cutoff function can "smooth" the series to arrive at − + 1 / 12 . Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler–Maclaurin formula.
The method of infinite descent is a variation of mathematical induction which was used by Pierre de Fermat. It is used to show that some statement Q(n) is false for all natural numbers n. Its traditional form consists of showing that if Q(n) is true for some natural number n, it also holds for some strictly smaller natural number m.
In these limits, the infinitesimal change is often denoted or .If () is differentiable at , (+) = ′ ().This is the definition of the derivative.All differentiation rules can also be reframed as rules involving limits.
Shakuntala Devi (4 November 1929 – 21 April 2013) was an Indian mental calculator, astrologer, and writer, popularly known as the "Human Computer".Her talent earned her a place in the 1982 edition of The Guinness Book of World Records.
Accordingly I distinguish an eternal uncreated infinity or absolutum, which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created ...
The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the ...