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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 Limits for general functions. Toggle Limits for general functions subsection. 1.1 Definitions of limits and related concepts. ... if n is a positive integer [1] [2] [3]
[2] [4] Oresme's work, and the contemporaneous work of Richard Swineshead on a different series, marked the first appearance of infinite series other than the geometric series in mathematics. [5] However, this achievement fell into obscurity. [6] Additional proofs were published in the 17th century by Pietro Mengoli [2] [7] and by Jacob Bernoulli.
2.1 Low-order polylogarithms. 2.2 Exponential function. 2.3 Trigonometric, inverse trigonometric, ... 7.2 Sum of reciprocal of factorials. 7.3 Trigonometry and ...
In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1.
where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value). If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
[2] [3] Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. [4] [5] The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. [6]
On one hand, the limit as n approaches infinity of a sequence {a n} is simply the limit at infinity of a function a(n) —defined on the natural numbers {n}. On the other hand, if X is the domain of a function f ( x ) and if the limit as n approaches infinity of f ( x n ) is L for every arbitrary sequence of points { x n } in X − x 0 which ...