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Concept
1 + 2 + 4 + 8 + ⋯
Summary
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).
Official source
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