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The n th partial sum of the series is the triangular number. Because the sequence of partial sums fails to converge Kusine - Psychoterror - Psychoterror a finite limitthe series does not have a sum. Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of 3 interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series.
These methods have applications in other fields such as complex analysisquantum field theoryand string theory. In a monograph on moonshine theoryTerry Gannon calls this equation "one of the most remarkable formulae in science".
The n th partial sum is given by a simple formula:. This equation was known to the Pythagoreans as early as the sixth century BCE. The divergence is a simple consequence of the form of the series: the terms 1 not approach zero, so the series diverges by the term test. Many summation methods are used to assign numerical values to divergent series, some more powerful than others.
More advanced methods are required, such as zeta function regularization or Ramanujan summation. The latter 2 is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century. These relationships can be expressed using algebra.
Then multiply this equation by 4 and subtract the second equation from the first:. Accordingly, Ramanujan writes:. Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums.
For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. For an extreme example, appending a single zero to the front 1 the series can lead to inconsistent results. One way to remedy this situation, and to 3 the places where zeroes may be inserted, is to keep 2 of each term in the series by attaching a dependence on some function.
The implementation of this strategy is called zeta function regularization. The latter series is an example of a Dirichlet series. The benefit of introducing the Riemann zeta function is that 1 can be defined for other values of s by analytic continuation.
The eta function is defined by an alternating Dirichlet series, so this Fuck Your Drama - The Rites - Wish You Never Knew parallels the earlier heuristics. Where both Dirichlet series converge, one has the identities:.
Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysisand Ramanujan summation, with its shortcut to the Euler—Maclaurin formula. Instead, the method operates directly on conservative transformations of the series, using methods from real analysis.
The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows. For convenience, one may require Beatles - Various - Svensktoppar Vol 30 f is smoothboundedand compactly supported.
Ramanujan wrote in his second letter to G. Hardydated 27 February Ramanujan summation is a method to isolate the constant term in the Euler—Maclaurin formula for the partial sums of a series. To avoid inconsistencies, the modern theory of Ramanujan summation requires that f is "regular" in the sense that the higher-order derivatives of f decay quickly enough for the remainder terms in the Euler—Maclaurin formula to tend to 0.
Ramanujan tacitly assumed this property. Instead, such a series must be interpreted by zeta 4 - Various - Energy 2013 regularization. For this reason, Hardy recommends 3 caution" when applying the Ramanujan sums of known series to find the sums of related series. Stable means that adding a term to the beginning of the series increases the sum by the same amount. This can be seen as follows. By linearity, one may subtract the second equation from the first subtracting each component of the second line from the first line in columns to give.
In bosonic string theorythe attempt is to compute the possible 4 - Various - Energy 2013 levels of a string, in particular the lowest energy level. Ultimately it is this 1, combined with the Goddard—Thorn theoremwhich leads to bosonic string theory failing to be consistent in dimensions other than The spatial symmetry 3 the problem is responsible for canceling the quadratic term of the expansion.
A similar calculation is involved in three dimensions, using the Epstein zeta-function in place of the Riemann zeta function. David Leavitt 's novel The Indian Clerk includes a scene where Hardy and Littlewood discuss the meaning of this series.
As Ruth launches into a derivation of the functional equation of the zeta function, another actor addresses the audience, admitting that they are actors: "But the mathematics is 3. It's terrifying, but it's real. In JanuaryNumberphile produced a YouTube video on the series, which gathered over 1. 3 The New York 3 coverage of the Numberphile video, mathematician Edward Frenkel commented, "This calculation is one of the best-kept secrets in math.
No one on the outside knows about it. From Wikipedia, the free encyclopedia. Main article: Triangular number. Misra ed. The Euler Archive. 3 Originally published as Euler, Leonhard Theory and Application of Infinite Series. May"Euler's paper on divergent series", Historia Mathematica3 2 : —, doi : Berndt, Bruce C. Elephants - 23 Skidoo - Just Like Everybody letters and commentary.
American Mathematical Society. Hardy, G. Divergent Series. Clarendon Press. Zee, A. Quantum field theory in a nutshell. Princeton UP. Zwiebach, Barton A First Course in String Theory. Cambridge UP. See p. Elizalde, Emilio Bibcode : gr. Watson, G. Sequences and series.
Cauchy sequence Monotone sequence Periodic sequence. Convergent series Divergent series Conditional convergence Absolute convergence Uniform convergence Alternating series Telescoping series. Riemann zeta function. Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Theta hypergeometric 1. Book Category.
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