1 + 2 + 3 + 4 + ⋯

Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of different mathematical results.

For example, many summation methods are used in mathematics to assign numerical values even to a divergent series.

In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of ⁠−+1/12⁠, which is expressed by a famous formula:[2]

where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning.

[3] In a monograph on moonshine theory, University of Alberta mathematician Terry Gannon calls this equation "one of the most remarkable formulae in science".

Many summation methods are used to assign numerical values to divergent series, some more powerful than others.

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.

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.

[1] 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.

When the real part of s is greater than 1, the Dirichlet series converges, and its sum is the Riemann zeta function ζ(s).

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.

Now, computing η(−1) is an easier task, as the eta function is equal to the Abel sum of its defining series,[13] which is a one-sided limit:

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.

I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter.

[14]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.

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 26.

[19] An exponential cutoff function suffices to smooth the series, representing the fact that arbitrarily high-energy modes are not blocked by the conducting plates.

All that is left is the constant term −1/12, and the negative sign of this result reflects the fact that the Casimir force is attractive.

[20] A similar calculation is involved in three dimensions, using the Epstein zeta-function in place of the Riemann zeta function.

According to Morris Kline, Euler's early work on divergent series relied on function expansions, from which he concluded 1 + 2 + 3 + 4 + ⋯ = ∞.

[27] David Leavitt's 2007 novel The Indian Clerk includes a scene where Hardy and Littlewood discuss the meaning of this series.

[28] Simon McBurney's 2007 play A Disappearing Number focuses on the series in the opening scene.

The main character, Ruth, walks into a lecture hall and introduces the idea of a divergent series before proclaiming, "I'm going to show you something really thrilling", namely 1 + 2 + 3 + 4 + ⋯ = ⁠−+1/12⁠.

"[29][30] In January 2014, Numberphile produced a YouTube video on the series, which gathered over 1.5 million views in its first month.

[32][full citation needed] Numberphile also released a 21-minute version of the video featuring Nottingham physicist Ed Copeland, who describes in more detail how 1 − 2 + 3 − 4 + ⋯ = ⁠1/4⁠ as an Abel sum, and 1 + 2 + 3 + 4 + ⋯ = ⁠−+1/12⁠ as ζ(−1).

[33][full citation needed] After receiving complaints about the lack of rigour in the first video, Padilla also wrote an explanation on his webpage relating the manipulations in the video to identities between the analytic continuations of the relevant Dirichlet series.

[34] In The New York Times coverage of the Numberphile video, mathematician Edward Frenkel commented: "This calculation is one of the best-kept secrets in math.

"[31] Coverage of this topic in Smithsonian magazine describes the Numberphile video as misleading and notes that the interpretation of the sum as ⁠−+1/12⁠ relies on a specialized meaning for the equals sign, from the techniques of analytic continuation, in which equals means is associated with.