"Uniform and Absolute Convergence"

Question

Marsden and Hoffman's Basic Complex Analysis describes some series of functions $\sum g_n : A \to \mathbb{C}$ as "converging uniformly and absolutely" to a function $g : A \to \mathbb{C}$. (An example is the conclusion of the Weierstrass $M$-test.) There's a small ambiguity here. Am I meant to interpret this as (1) $\sum g_n \to g$ uniformly and $g_n$ converges absolutely pointwise? Or is it (2) $\sum |g_n|$ converges uniformly and $g_n \to g$? In all instances so far in the text, (2) is satisfied, and certainly (2) implies (1). But (1) feels like a more literal translation of the phrase "converges uniformly and absolutely." Which definition should I go for?

Answer 1

I would interpret it as $\sum_{n}g_n$ converges to $g$ uniformly, and $\sum_{n}|g_n(x)| < \infty$ for all $x$. So at each point, the series converges absolutely. And the series converges to $g$ uniformly.

Answer 2

The expression "$\sum u_n$ converges uniformly and absolutely" should be considered as a shortcut for "$\sum u_n$ converges uniformly and $\sum u_n$ converges absolutely", ie. "$\sum u_n$ converges uniformly and $\sum |u_n|$ converges pointwise" (already pointed out by Mason). This is what we call uniform and absolute convergence.

The other possibility, ie. "$\sum |u_n|$ converges uniformly", might be called uniform absolute convergence but I do not claim this is a standard terminology.

An interesting question naturally arises: are these two notions of convergence equivalent or not? Obviously, the latter implies the former. But the converse is not true. In fact, Weierstrass M-test is a test of the latter and it is a sufficient but not necessary condition for the former.