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Secant numbers are related to the signed Euler numbers (Taylor coefficients of hyperbolic secant) by the formula E 2n = (−1) n A 2n. (E n = 0 when n is odd.) Correspondingly, the numbers A 2n+1 with odd indices are called tangent numbers or zag numbers. The first few values are 1, 2, 16, 272, 7936, ... (sequence A000182 in the OEIS).
In mathematics, an alternating series is an infinite series of terms that alternate between positive and negative signs. In capital-sigma notation this is expressed ∑ n = 0 ∞ ( − 1 ) n a n {\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}} or ∑ n = 0 ∞ ( − 1 ) n + 1 a n {\displaystyle \sum _{n=0}^{\infty }(-1)^{n+1}a_{n}} with a n ...
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: = = + + + + +.. The first terms of the series sum to approximately +, where is the natural logarithm and is the Euler–Mascheroni constant.
7.1 Alternating harmonic series. ... is a Bernoulli number, and here, = . is an Euler number ... See Faulhaber's formula.
The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate π to 5,263 decimal places with the Leibniz ...
The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. The latter is the function in the definition. They also occur in combinatorics , specifically when counting the number of alternating permutations of a set with an even number of elements.
In mathematics, Faulhaber's formula, ... and with alternating signs. Let ... call number QA154.8 F3 1631a f MATH.
In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test , Leibniz's rule , or the Leibniz criterion .