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 lacks a sum in the usual sense. On the other hand, its Cesàro sum is 1/2.
One obvious method to attack 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 we may use 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,
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. 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:
In fact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts that arose in the 19th century. After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians.
In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. The sequence of partial sums of Grandi's series is 1, 0, 1, 0, …, which clearly does not approach any number (although it does have two accumulation points at 0 and 1). Therefore, Grandi's series is divergent.
It can be shown that it is not valid to perform many seemingly innocuous operations on a series, such as reordering individual terms, unless the series is absolutely convergent. Otherwise these operations can alter the result of summation. It's easy to see how terms of Grandi's series can be rearranged to arrive at any integer number, not only 0 or 1.
- E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series (Cambridge University Press, 1907), section 331. The University of Michigan Historical Mathematics Collection 
- E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th edition, reprinted (Cambridge University Press, 1962), section 2.1.
- Devlin p.77
- Davis p.152
- Kline 1983 p.307
- Knopp p.457
- Davis, Harry F. (May 1989). Fourier Series and Orthogonal Functions. Dover. ISBN 0-486-65973-9.
- Devlin, Keith (1994). Mathematics, the science of patterns: the search for order in life, mind, and the universe. Scientific American Library. ISBN 0-7167-6022-3.
- Kline, Morris (November 1983). "Euler and Infinite Series". Mathematics Magazine 56 (5): 307–314. doi:10.2307/2690371. JSTOR 2690371.
- Knopp, Konrad (1990) . Theory and Application of Infinite Series. Dover. ISBN 0-486-66165-2.