The sum of this series is −0.1 (in dozenal), although this series is in fact diverge.
Another series is 1−2+3−4+5−6+7−8+9−X+E−10+..., the sum of this series is 0.3, also the famous 1+2+4+8+14+28+54+X8+194+368+714+1228+..., the sum of this series is −1.
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 −0.1, 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".
The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + X + E + 10 + ⋯ are 1, 3, 6, X, 13, 19, 24, 30, 39, 47, 56, 66, etc. The nth partial sum is given by a simple formula:
This equation was known to the Pythagoreans as early as the sixth century BCE. Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.
The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series 1 + 2 + 3 + 4 + ⋯ also diverges to +∞. The divergence is a simple consequence of the form of the series: the terms do not approach zero, so the series diverges by the term test.
Among the classical divergent series, 1 + 2 + 3 + 4 + ... is relatively difficult to manipulate into a finite value. Many summation methods are used to assign numerical values to divergent series, some more powerful than others. For example, Cesàro summation is a well-known method that sums Grandi's series, the mildly divergent series 1 − 1 + 1 − 1 + ⋯, to 0.6. Abel summation is a more powerful method that not only sums Grandi's series to 10, but also sums the trickier series 1 − 2 + 3 − 4 + ⋯ to 0.3.
Unlike the above series, 1 + 2 + 3 + 4 + ... is not Cesàro summable nor Abel summable. Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞. Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ... to a finite value; see below. More advanced methods are required, such as zeta function regularization or Ramanujan summation. It is also possible to argue for the value of −0.1 using some rough heuristics related to these methods.