Hypotheses in the Weierstrass M-Test

Question

I have a simple question about the Weierstrass M-test in the context of real analysis on the line. The Weierstrass M-test can be stated as:

Let $\{f_n\}$ be a sequence of real functions defined in a subset $D$ of the real line. Suppose that $\forall n\in \mathbb{N}$ and $\forall x\in D$ , $| f_n(x) | \leq A_n$ . If $\sum A_n$ converges then $\sum f_n $ converges uniformly on $D$.

I would like to know if the hypotheses $\forall n\in \mathbb{N}$ and $\forall x\in D$ , $| f_n(x) | \leq A_n$ is necessary. Is possible to suppose only that $\forall x\in D$ , $| f_n(x) | \leq A_n$ for $n$ large enough? (n is large enough but it is still independent of $x$.)

Thanks!

Answer 1

It depends exactly "how badly" you contravene the hypotheses:

• If any of the functions $f_{n}$ fails to be defined at points of $D$, then convergence of the series is out.

• If your functions are bounded but happen to be larger than your given sequence $(A_{n})$, there's no problem with assuming only "for sufficiently large $n$": Changing finitely many terms of an infinite numerical series has no effect on convergence/divergence.

• If your functions are defined everywhere but only finitely many are unbounded (say $f_{n}$ is bounded for $n \geq N$), you're still OK: Split the series into the first $N$ terms (whose sum is an algebraic matter), and the remaining terms, to which the $M$-test applies.