1-1+1-1+1-1+1... What is the value of this infinite sequence?

The infinite series 1 − 1 + 1 − 1 + 1 − 1 + … has no single convergent value in the standard framework of classical analysis; it is a canonical example of a divergent series. Its partial sums oscillate perpetually between 1 (when summing an odd number of terms) and 0 (when summing an even number of terms), never approaching a finite limit. Therefore, within the conventional definition of an infinite series as the limit of its partial sums, this series does not sum to any real number. This simple pattern, historically known as Grandi's series after the 18th-century mathematician Guido Grandi, has played a surprisingly profound role in the development of summation theory by highlighting the limitations of traditional limits and necessitating more sophisticated conceptual tools.

The intrigue and historical debate around this series stem from the fact that one can produce seemingly plausible arguments for different finite sums, most commonly 0, 1, or 1/2. For instance, grouping the terms as (1 − 1) + (1 − 1) + … suggests a sum of 0, while rewriting it as 1 + (−1 + 1) + (−1 + 1) + … suggests a sum of 1. These manipulations are invalid for a divergent series because they rely on the false assumption of associativity for infinite sums without convergence. The more interesting candidate, 1/2, emerges when one applies specific generalized summation methods designed to assign meaningful values to otherwise divergent series. Notably, Cesàro summation, which averages the partial sums, yields a Cesàro sum of 1/2, as the sequence of arithmetic means of the oscillating partial sums (1, 1/2, 2/3, 1/2, 3/5, …) converges to 1/2.

This series is not merely a mathematical curiosity; it serves as a critical gateway to understanding the broader philosophy and utility of summation methods. Techniques like Abel summation, which considers the limit of the power series Σ(−1)^n x^n as x approaches 1 from below, also yield a value of 1/2, as this generating function 1/(1+x) tends to 1/2. These consistent results from different regularization methods indicate that 1/2 can be a useful and stable *assigned value* in specific analytic contexts, such as Fourier analysis or theoretical physics, where divergent series formally arise. However, it is crucial to maintain the distinction: the series is divergent in the ordinary sense, but 1/2 is its Cesàro sum or Abel sum.

The ultimate answer to the question of its value is thus context-dependent. In elementary calculus and standard analysis, the correct answer is that the series diverges. In more advanced studies of asymptotic series or regularization techniques, one may legitimately assign it the value 1/2 for specific operational purposes. This duality underscores a fundamental principle: the "sum" of an infinite series is defined by the summation method one employs, and no single method holds absolute priority. The series 1 − 1 + 1 − 1 + … therefore stands as a permanent reminder that the concept of infinity in mathematics requires carefully specified rules of engagement.