page_title: 1 + 2 + 3 + 4 + ⋯
section_title: Heuristics
hierarchical_section_title: 1 + 2 + 3 + 4 + ⋯ / Summability / Heuristics
caption_attribution_description: English: Snippet from Srinivasa Ramanujan's first notebook, chapter 8, concerning an alternate derivation of 1 + 2 + 3 + 4 + ⋯ = −1/12
context_page_description: The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number
  which increases without bound as n goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the 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 mathematically interesting 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 −+, which is expressed by a famous formula,
  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. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory.
  In a monograph on moonshine theory, Terry Gannon calls this equation "one of the most remarkable formulae in science".
context_section_description: Srinivasa Ramanujan presented two derivations of "1 + 2 + 3 + 4 + ⋯ = −+" in chapter 8 of his first notebook. The simpler, less rigorous derivation proceeds in two steps, as follows.
  The first key insight is that the series of positive numbers 1 + 2 + 3 + 4 + ⋯ closely resembles the alternating series 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.
  In order to transform the series 1 + 2 + 3 + 4 + ⋯ into 1 − 2 + 3 − 4 + ⋯, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is 4 + 8 + 12 + 16 + ⋯, which is 4 times the original series. These relationships can be expressed using algebra. Whatever the "sum" of the series might be, call it c = 1 + 2 + 3 + 4 + ⋯. Then multiply this equation by 4 and subtract the second equation from the first:
  The second key insight is that the alternating series 1 − 2 + 3 − 4 + ⋯ is the formal power series expansion of the function  but with x defined as 1. Accordingly, Ramanujan writes:
  Dividing both sides by −3, one gets c = −+.
  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. In particular, the step 4c = 0 + 4 + 0 + 8 + ⋯ is not justified by the additive identity law alone. 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. In the series 1 + 2 + 3 + 4 + ⋯, each term n is just a number. If the term n is promoted to a function n⁻ˢ, where s is a complex variable, then one can ensure that only like terms are added. The resulting series may be manipulated in a more rigorous fashion, and the variable s can be set to −1 later. The implementation of this strategy is called zeta function regularization.
caption: Passage from Ramanujan's first notebook describing the "constant" of the series
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