Weierstrass M-Test

Theorem
Suppose:


 * $$f_{n}$$ is a sequence of functions defined on a domain $$D$$;


 * $$ \sup_{x\in D} |f_{n}(x)| \leq M_{n}$$ for each integer $$n$$ and some constants $$M_{n}$$;


 * $$\sum_{i=1}^{\infty}M_{i} < \infty$$.

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

Proof
Let $$S_{n} = \sum_{i=1}^{\infty}f_{i}$$, and let $$f = \lim_{n\to \infty}S_{n}$$.

To show the partial sums converge uniformly to $$f$$, we must show that $$\lim_{n\to\infty}\sup_{x\in D} |f - S_{n}| = 0$$.

But $$\sup_{x\in D} |f - S_{n}| = \sup_{x\in D} |(f_{1} + f_{2} + ...) - (f_{1} + f_{2} + ... + f_{n})| = \sup_{x\in D} |f_{n+1} + f_{n+2} + \ldots|$$.

By the Triangle Inequality, this value is less than or equal to $$\sum_{i=n+1}^{\infty}\sup_{x\in D}|f_{i}(x)| \leq \sum_{i=n+1}^{\infty}M_{i}$$.

But since $$0 \leq \sum_{i=1}^{\infty}M_{n} < \infty$$, and a convergent series has tails that converge to zero, it follows that:
 * $$0 \leq \lim_{n\to\infty}\sum_{i=n+1}^{\infty}\sup_{x\in D}|f_{i}(x)| \leq \lim_{n\to\infty}\sum_{i=n+1}^{\infty}M_{i} = 0$$

So $$\lim_{n\to\infty}\sup_{x\in D}|f - S_{n}| = 0$$.

Hence the series converges uniformly on the domain.