Recall that a series of functions is an infinite sum of functions, and that the series converges when this infinite sum has a finite value. A simple example is the series 1 + x + *x 2 + *x 3 + … (the sum of the positive integer powers of x), which, for every x between -1 and 1, converges to 1/(1–x).
Besides the convergence of expressions of this kind, there is the question of how regular the resulting sum is. Cauchy believed that if the functions being summed were continuous, their sum—provided it converged—had to be continuous as well.
In 1826, Abel pointed out a counterexample: the series
converges but is not continuous at π, 3π, 5π, and so on.
Incidentally, Abel in turn made an error in the same article—but one with less serious consequences.
