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

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?

• If you've shown $|f_k|\le M_k$ and $\sum M_k<\infty$, then you've already shown it: the series converges absolutely everywhere by the comparison test. – zhw. Dec 19 '17 at 18:53

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.