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Concept
Grandi's series
Summary
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. For example, the Cesàro summation and the Ramanujan summation of this series is 1/2.
One obvious method to find the sum of the series
1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + ...
is to treat it like a telescoping series and perform the subtractions in place:
(1 − 1) + (1 − 1) + (1 − 1) + ... = 0 + 0 + 0 + ... = 0.
On the other hand, a similar bracketing procedure leads to the apparently contradictory result
1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... = 1 + 0 + 0 + 0 + ... = 1.
Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value". (Variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebra.)
Treating Grandi's series as a divergent geometric series and using the same algebraic methods that evaluate convergent geometric series to obtain a third value:
S = 1 − 1 + 1 − 1 + ..., so
1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S
1 − S = S
1 = 2S,
resulting in S = 1/2. The same conclusion results from calculating −S, subtracting the result from S, and solving 2S = 1.
The above manipulations do not consider what the sum of a series actually means and how said algebraic methods can be applied to divergent geometric series. Still, to the extent that it is important to be able to bracket series at will, and that it is more important to be able to perform arithmetic with them, one can arrive at two conclusions:
The series 1 − 1 + 1 − 1 + ... has no sum.
but its sum should be 1/2.
Official source
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