Weierstrass M-Test

Theorem
Let $f_n$ be a sequence of real functions defined on a domain $D \subseteq \R$.

Let $\displaystyle \sup_{x \mathop \in D} \size {\map {f_n} x} \le M_n$ for each integer $n$ and some constants $M_n$

Let $\displaystyle \sum_{i \mathop = 1}^\infty M_i < \infty$.

Then $\displaystyle \sum_{i \mathop = 1}^\infty f_i$ converges uniformly on $D$.

Proof
Let:
 * $\displaystyle S_n = \sum_{i \mathop = 1}^n f_i$

Let:
 * $\displaystyle f = \lim_{n \mathop \to \infty} S_n$

To show the partial sums converge uniformly to $f$, we must show that:
 * $\displaystyle \lim_{n \mathop \to \infty} \sup_{x \mathop \in D} \size {f - S_n} = 0$

But:

By the Triangle Inequality, this value is less than or equal to:
 * $\displaystyle \sum_{i \mathop = n + 1}^\infty \sup_{x \mathop \in D} \size {\map {f_i} x} \le \sum_{i \mathop = n + 1}^\infty M_i$

We have that:
 * $\displaystyle 0 \le \sum_{i \mathop = 1}^\infty M_n < \infty$

It follows from Tail of Convergent Series tends to Zero:
 * $\displaystyle 0 \le \lim_{n \mathop \to \infty} \sum_{i \mathop = n + 1}^\infty \sup_{x \mathop \in D} \size {\map {f_i} x} \le \lim_{n \mathop \to \infty} \sum_{i \mathop = n + 1}^\infty M_i = 0$

So:
 * $\displaystyle \lim_{n \mathop \to \infty} \sup_{x \mathop \in D} \size {f - S_n} = 0$

Hence the series converges uniformly on the domain.

Also known as
Some sources do not use the hyphen: Weierstrass $M$ test.