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 converges uniformly to a sum
if the sequence of partial sums
converges uniformly to
. That is, if for every
there is an
so that
implies that
for all
in the domain under consideration.
And we’ve got Cauchy’s condition: a series converges uniformly if for every there is an
so that
and
both greater than zero implies that
for all
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: for all
in the domain. And further assume that the series
converges. Then the series
must converge uniformly.
Since the series of the converges, Cauchy’s condition for series of numbers tells us that for every
there is some
so that when
and
are bigger than
,
. But now when we consider
we note that it’s just a finite sum, and so we can use the triangle inequality to write
So Cauchy’s condition tells us that the series converges uniformly in the domain under consideration.
[…] 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 […]
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