The Unapologetic Mathematician

Mathematics for the interested outsider

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.

About these ads

September 9, 2008 - Posted by | Analysis, Calculus, Functional Analysis

2 Comments »

  1. [...] 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

  2. 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


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 392 other followers

%d bloggers like this: