Weierstrass M-Test

Theorem

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

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

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

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

Proof

Let:

$\ds S_n = \sum_{i \mathop = 1}^n f_i$

Let:

$\ds f = \lim_{n \mathop \to \infty} S_n$

To show the sequence of partial sums converge uniformly to $f$, we must show that:

$\ds \lim_{n \mathop \to \infty} \sup_{x \mathop \in D} \size {f - S_n} = 0$

But:

$\ds \sup_{x \mathop \in D} \size {f - S_n}$

$=$

$\ds \sup_{x \mathop \in D} \size {\paren {f_1 + f_2 + \dotsb} - \paren {f_1 + f_2 + \dotsb + f_n} }$

$\ds $

$=$

$\ds \sup_{x \mathop \in D} \size {f_{n + 1} + f_{n + 2} + \dotsc}$

By the Triangle Inequality, this value is less than or equal to:

$\ds \sum_{i \mathop = n + 1}^\infty \sup_{x \mathop \in D} \size {\map {f_i} x} \le \sum_{i \mathop = n + 1}^\infty M_i$

We have that:

$\ds 0 \le \sum_{i \mathop = 1}^\infty M_n < \infty$

It follows from Tail of Convergent Series tends to Zero:

$\ds 0 \le \lim_{n \mathop \to \infty} \sum_{i \mathop = n + 1}^\infty \sup_{x \mathop \in D} \size {\map {f_i} x} \le \lim_{n \mathop \to \infty} \sum_{i \mathop = n + 1}^\infty M_i = 0$

So:

$\ds \lim_{n \mathop \to \infty} \sup_{x \mathop \in D} \size {f - S_n} = 0$

Hence the series converges uniformly on the domain.

$\blacksquare$

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Also known as

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

Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.

Historical Note

The was developed by Karl Weierstrass during his investigation of power series.

Sources

1973: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): $\S 9.6$: Uniform Convergence of Infinite Series of Functions: Theorem $9.6$
1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.3$: Infinite series of functions: Theorem $1.8$