1 + 2 + 4 + 8 + ⋯

In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so the sum of this series is infinity. However, 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. For example, the Ramanujan summation of this series is −1, which is the limit of the series using the 2-adic metric. The partial sums of are since these diverge to infinity, so does the series. Therefore, any totally regular summation method gives a sum of infinity, including the Cesàro sum and Abel sum. On the other hand, there is at least one generally useful method that sums to the finite value of −1. The associated power series has a radius of convergence around 0 of only so it does not converge at Nonetheless, the so-defined function has a unique analytic continuation to the complex plane with the point deleted, and it is given by the same rule Since the original series is said to be summable (E) to −1, and −1 is the (E) sum of the series. (The notation is due to G. H. Hardy in reference to Leonhard Euler's approach to divergent series). An almost identical approach (the one taken by Euler himself) is to consider the power series whose coefficients are all 1, that is, and plugging in These two series are related by the substitution The fact that (E) summation assigns a finite value to shows that the general method is not totally regular. On the other hand, it possesses some other desirable qualities for a summation method, including stability and linearity. These latter two axioms actually force the sum to be −1, since they make the following manipulation valid: In a useful sense, is a root of the equation (For example, is one of the two fixed points of the Möbius transformation on the Riemann sphere).

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Grandi's series

In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯, also written is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it does not have a sum. However, 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.

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Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. Since there are no properties of an entire sum, the Ramanujan summation functions as a property of partial sums.

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