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1 + 2 + 3 + 4 + ⋯

Divergent series / From Wikipedia, the free encyclopedia

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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

A graph depicting the series with layered boxes and a parabola that dips just below the y-axis
The first four partial sums of the series 1 + 2 + 3 + 4 + ⋯. The parabola is their smoothed asymptote; its y-intercept is −1/12.[1]

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 +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. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory.[3]

In a monograph on moonshine theory, University of Alberta mathematician Terry Gannon calls this equation "one of the most remarkable formulae in science".[4]

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