How to understand whether Weierstrass $M$ test is applicable or not.

Question

Suppose I am given a series of function $\sum f_n(x)$ on the domain $X$. Now it may happen that its uniform convergence cannot be proved using $M$ test. How do I understand that whether the test is applicable or I have to look for some other alternative. How to prove that for a given series I cannot use this Weierstrass test to show its uniform convergence, is there any rule of hand that will guarantee that we cannot prove its uniform convergence with the help of $M$ test?

Answer 1

If you are able to compute $s_n=\sup_x\bigl\lvert f_n(x)\bigr\rvert$ for each $n\in\mathbb N$, then you can apply the Weierstrass $M$-test if and only if the series $\sum_{n=1}^\infty s_n$ converges. This is so because:

• if there is a sequence $(M_n)_{n\in\mathbb N}$ of non-negative numbers such that the series $\sum_{n=1}^\infty M_n$ converges and that$$(\forall x\in X)(\forall n\in\mathbb N):\bigl\lvert f_n(x)\bigr\rvert\leqslant M_n,\tag1$$then $(\forall n\in\mathbb N):s_n\leqslant M_n$ and therefore, by the comparison test, the series $\sum_{n=1}^\infty s_n$ converges;
• if, for every sequence $(M_n)_{n\in\mathbb N}$ of non-negative numbers such $(1)$ holds, the series $\sum_{n=1}^\infty M_n$ diverges, then, since the sequence $(s_n)_{n\in\mathbb N}$ is one such sequence, the series $\sum_{n=1}^\infty s_n$ also diverges.