Do I need to check if a series converges absolutely first to be able to use Weierstrass' M-test?

Question

The Weierstrass M-Test states that if you have a sequence of functions $(f_k)_{k\epsilon\mathbb N}$ where $f_k:A\mapsto\mathbb R$, and suppose that $\forall k \epsilon \mathbb N \exists M_k >0 $ such that

$$|f_k(x)|\leq M_k \forall x \epsilon A$$ and $$\sum_{k=1}^{\infty}M_k <\infty$$

Then the series $\sum_{k}f_k$ converges uniformly on A.

However, the Weierstrass M-test is not applicable to series that converge uniformly but not absolutely.

Therefore, when attempting to prove a series converges uniformly using the M test, do I need to show first of all that the series converges absolutely?

Answer 1

Not really. The conclusion of the usual statement of the M test (cf Wikipedia) is that the series converges uniformly and absolutely. So obviously if it does not converge absolutely you will never be able to use the M test, but that will be reflected in the premises of the M test not holding.

In particular, if $f_n(x)$ did not converge absolutely for some $x,$ and the premises of the m test held we would have $$\infty=\sum_n |f_n(x)| \leq \sum_n M_n < \infty,$$ a clear contradiction.