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In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
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A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
0 (zero) is a number representing an empty quantity.Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures.
In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power.Exponentiation is written as b n, where b is the base and n is the power; often said as "b to the power n ". [1]
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2]
[1]: 124 For example, when determining a derivative of a function the following formula is used: Q ( h ) = f ( a + h ) − f ( a ) h . {\displaystyle Q(h)={\frac {f(a+h)-f(a)}{h}}.} Intuitively one would want an h very close to zero; however, when using floating-point operations, the smallest number will not give the best approximation of a ...
If it can be proved that if a decimal of the form ... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem. [47] Investigations in this direction can motivate such concepts as greatest common divisors , modular arithmetic , Fermat primes , order of group elements, and quadratic reciprocity .