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.)

Written by

François Apéry

PublishedFebruary 22, 2023UpdatedSeptember 19, 20261 min read

In everyday life, we deal only with finite sums:

2 + 2 = 4, 1³ + 2³ + 3³ + 4³ = (1 + 2 + 3 + 4)².

A priori, an infinite sum is meaningless. Why, then, do mathematicians go to such lengths to manipulate infinite sums? The longing for infinity is inherent in the human soul and has found expression, at different times, through religion, philosophy, art and science.

Infinity and its paradoxes -------------------------

At a conference in 1880, the French mathematician Fortuné Landry (1799–1895) steps up to the board and writes:

2^{2^6} +1 = 274177 \times 67 280421310721.

Discussion

Comments

Join the discussion

No comments yet. Be the first to respond.

On the same theme

Articles recommended for you.

History and Culture

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.

François ApéryFeb 23, 2023

History and Culture

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.

BERTRAND HAUCHECORNEFeb 22, 2023

Math for everyone

Timeless mathematical puzzles | Tangente

Puzzles have been part of human culture since earliest antiquity. At first, inventing puzzles was bound up with mythology or religion; gradually, it became a purely intellectual game, independent of any purpose or practical application.

MICHEL CRITONJun 10, 2021

Games and Challenges

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.

Jacques BairFeb 22, 2023