1, 2, 3, 4

The latter series 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:. Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums.

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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 of the series can lead to inconsistent results. One way to remedy this situation, and to constrain the places where zeroes may be inserted, is to keep track of each term in the series by attaching a dependence on some function.

Can the sum of all positive integers = -1/12? It can, sort of…

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 it can be defined for other values of s by analytic continuation. The eta function is defined by an alternating Dirichlet series, so this method 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 analysis , and 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 that f is smooth , bounded , and compactly supported. The constant term of the asymptotic expansion does not depend on f: Ramanujan wrote in his second letter to G.

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Hardy , dated 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.

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Ramanujan tacitly assumed this property. Instead, such a series must be interpreted by zeta function regularization. For this reason, Hardy recommends "great 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 theory , the attempt is to compute the possible energy levels of a string, in particular the lowest energy level.

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Ultimately it is this fact, combined with the Goddard—Thorn theorem , which leads to bosonic string theory failing to be consistent in dimensions other than The spatial symmetry of 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: It's terrifying, but it's real.

In January , Numberphile produced a YouTube video on the series, which gathered over 1. Divergent series are on the whole devil's work, and it is a shame that one dares to found any proof on them. One can get out of them what one wants if one uses them, and it is they which have made so much unhappiness and so many paradoxes.

Can one think of anything more appalling than to say that. Here's something to laugh at, friends. The series are also studied for non-integer values of n ; these make up the Dirichlet eta function.

What is the sum of the first whole numbers?

The eta function in particular is easier to deal with by Euler's methods because its Dirichlet series is Abel summable everywhere; the zeta function's Dirichlet series is much harder to sum where it diverges. From Wikipedia, the free encyclopedia. For the full details of the calculation, see Weidlich, pp. Ferraro criticizes Tucciarone's explanation p. Although the paper was written in , it was not published until Euler's advice is vague; see Euler et al. John Baez even suggests a category-theoretic method involving multiply pointed sets and the quantum harmonic oscillator.

Archived at the Wayback Machine. Retrieved on March 11, Fourier Series and Orthogonal Functions. Remarks on a beautiful relation between direct as well as reciprocal power series". Originally published as Euler, Leonhard Ferraro, Giovanni June An Aspect of the Rise of 20th Century Mathematics". Archive for History of Exact Sciences. The development of the foundations of mathematical analysis from Euler to Riemann. Kline, Morris November Author also known as A.

1 + 2 + 3 + 4 + ⋯

Distributions in the Physical and Engineering Sciences, Volume 1. Tucciarone, John January Fourier Analysis and Its Applications. Summability methods for divergent series.

Cauchy sequence Monotone sequence Periodic sequence. Convergent series Divergent series Conditional convergence Absolute convergence Uniform convergence Alternating series Telescoping series. Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Theta hypergeometric series. Retrieved from " https: