Since series of anything are special cases of sequences, we can import our notions to series. We say that a series converges uniformly to a sum if the sequence of partial sums converges uniformly to . That is, if for every there is an so that implies that for all in the domain under consideration.
And we’ve got Cauchy’s condition: a series converges uniformly if for every there is an so that and both greater than zero implies that for all in the domain.
Here’s a a great way to put this to good use: the Weierstrass M-test, which is sort of like the comparison test. Say that we have a positive bound for the size of each term in the series: for all in the domain. And further assume that the series converges. Then the series must converge uniformly.
Since the series of the converges, Cauchy’s condition for series of numbers tells us that for every there is some so that when and are bigger than , . But now when we consider we note that it’s just a finite sum, and so we can use the triangle inequality to write
So Cauchy’s condition tells us that the series converges uniformly in the domain under consideration.