Summing series, even divergent ones
Infinite sequences of numbers have always puzzled mathematicians, who have consensually developed the notion of convergence and limit. But everything changed when they became interested in the successive sums of these sequences, to which they gave the name of series. Convergent series (the "nice" ones), studied by all those destined for a scientific career, pose no problem. They allow defining the sum of an infinite number of terms, as long as the "partial sums" converge to a limit. But the others, officially "divergent" (the "wicked" ones), were at the origin of very different, sometimes astonishing approaches, creating true philosophical conflicts between famous scientists. Euler, Abel and some pioneers ventured to imagine defining sums for them, finding some wonders… which today are the subject of a beautiful theory!
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A brief history of Grandi's series
Grandi's series is the infinite sum 1 – 1 + 1 – 1 + 1 – 1 + … Mathematicians have studied it for more than three hundred years. Its history can be divided into three stages.
Infinite sums: a matter of convention
Equality between two numbers poses no difficulty when both are defined by "finitary"* methods. But as soon as infinity enters the picture—represented by ellipses in formulas—the door opens to all manner of paradoxes. (* Finitism is an approach to mathematics that considers only finite objects.)
Converging to a number: is there only one way?
Traditional mathematics provides a precise definition of convergence for a numerical series, explored in the main body of this article. But this should not rule out less conventional approaches, discussed in the box and in several articles in this special issue.
The triumphs and tribulations of summing a series
Every convergent series can be assigned a sum—but what about divergent series? Mathematical orthodoxy holds that they cannot be assigned a value. Yet Leibniz and Euler suggested a few possible approaches. Poisson, Frobenius and Borel later crossed that forbidden line.
Assigning a value… to a divergent series!
Beyond the "classical" convergence of numerical series, many methods—some particularly powerful—can accelerate the convergence of a series or even assign a value to the "sum of a divergent series." Here are a few notable examples.