Uniform Convergence of Series

Since series of anything are special cases of sequences, we can import our notions to series. We say that a series $\sum\limits_{n=0}^\infty f_n$ converges uniformly to a sum $f$ if the sequence of partial sums $s_n=\sum\limits_{k=0}^nf_k$ converges uniformly to $f$ . That is, if for every $\epsilon>0$ there is an $N$ so that $n>N$ implies that $\left|f-\sum\limits_{k=0}^nf_k(x)\right|<\epsilon$ for all $x$ in the domain under consideration.

And we’ve got Cauchy’s condition: a series converges uniformly if for every $\epsilon>0$ there is an $N$ so that $m$ and $n$ both greater than zero implies that $\left|\sum\limits_{k=m}^nf_k(x)\right)<\epsilon$ for all $x$ in the domain.

Here’s a a great way to put this to good use: the Weierstrass M-test, which is sort of like the comparison test. Say that we have a positive bound for the size of each term in the series: $\left|f_n(x)\right|<M_n$ for all $x$ in the domain. And further assume that the series $\sum\limits_{n=0}^\infty M_n$ converges. Then the series $\sum\limits_{n=0}^\infty f_n(x)$ must converge uniformly.

Since the series of the $M_n$ converges, Cauchy’s condition for series of numbers tells us that for every $\epsilon>0$ there is some $N$ so that when $m$ and $n$ are bigger than $N$ , $\sum\limits_{k=m}^nM_n<\epsilon$ . But now when we consider $\left|\sum\limits_{k=m}^nf_n(x)\right|$ we note that it’s just a finite sum, and so we can use the triangle inequality to write

$\left|\sum\limits_{k=m}^nf_n(x)\right|\leq\sum\limits_{k=m}^n\left|f_n(x)\right|<\sum\limits_{k=m}^nM_n<\epsilon$

So Cauchy’s condition tells us that the series $\sum\limits_{n=0}^\infty f_n(x)$ converges uniformly in the domain under consideration.

2 Comments »

[…] some point so that for every point we have . And thus we have for all . Setting , we invoke the Weierstrass M-test — the series converges because is within the disk of convergence, and thus evaluation at […]

Pingback by Uniform Convergence of Power Series « The Unapologetic Mathematician | September 10, 2008 | Reply
Just wish to say your article is as amazing. The clearness for your submit is just excellent and i
could suppose you’re knowledgeable on this subject. Well together with your permission allow me to snatch your RSS feed to stay up to date with forthcoming post. Thanks 1,000,000 and please continue the enjoyable work.

Comment by pit online 2014 | July 27, 2013 | Reply

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

Recent Posts
Blogroll
Art
- Sketches of Topology
Astronomy
- Bad Astronomy
- Starts With a Bang
Computer Science
Education
Mathematics
Me
- DrMathochist
- The Unapologetic Programmer
Philosophy
Physics
Politics
Science

M	T	W	T	F	S	S
« Aug				Oct »
1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30

The Unapologetic Mathematician

Mathematics for the interested outsider