Weierstrass M-Test

From ProofWiki
Jump to navigation Jump to search

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.

$\blacksquare$


This article is complete as far as it goes, but it could do with expansion.
In particular: Establish how broadly this can be applied - it's defined for real functions but should also apply to complex ones and possibly a general metric space.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



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 Weierstrass M-Test was developed by Karl Weierstrass during his investigation of power series.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Weierstrass_M-Test&oldid=514247"