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.

September 9, 2008 Posted by | Analysis, Calculus, Functional Analysis | 2 Comments

   

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