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Transcript
Roberson 1
Lee Roberson
Complex Analysis
5/05/09
WEIERSTRASS M-TEST
Before proving the Weierstrass M-Test we will need to gather a few tools.
THEORM
(CAUCHY CRITERION FOR CONVERGENCE):
Suppose
for any
is a sequence of complex numbers for
there exists an
such that
.
. Then
converges if and only if
for every
such that
Any sequence that satisfies the Cauchy Criterion is known as a Cauchy sequence. The
above theorem also shows that every convergent sequence is Cauchy, and every Cauchy
sequence is convergent.
COROLLARY 1:
If
is a Cauchy sequence that converges to Z, and N is chosen such that
every
such that
, then for each
,
for
..
PROOF:
This proof is rather straightforward. Let
this that
.
in the inequality
. It follows from
COROLLARY 2:
The series
converges if and only if for any
for every
such that
there exists an N such that
.
Now it is possible to prove the generalized Comparison Test, also known as the
Weierstrass M-Test.
Roberson 2
THEOREM
(WEIERSTRASS M-TEST FOR CONVERGENCE OF FUNCTIONAL SERIES):
Suppose
is a sequence of real- or complex-valued functions on some set E. Also, suppose
that
is a convergent series where
are real non-negative terms. If
for
all k greater than some number N and for all z in some set E, then it follows that the series
converges uniformly on E.
PROOF:
Since
is Cauchy, we can choose a number
such that for any m and n that satisfy
we get that
. Then we see that for z in the set E that our series
is also Cauchy, since
Therefore,
function
converges for every
. Let us say that
converges to the
.
Now, we want to show that
rewrite
converges uniformly to
. Observe that we can
in terms of partial sums
for all
, and where
Theorem, we see that
for
, and where
. Then applying the 1st Corollary of the Cauchy Criterion
. Thus, the uniform convergence is shown.
Roberson 3
So now, if we look at the Comparison Test, we see that it is a special case of the
Weierstrass M-Test where
is just a constant function.
THEOREM
(COMPARISON TEST):
Suppose we have the terms
such that
Then if the series
converges, the series
for all
,
for some number N.
converges as well.
Since we know some of the ideas behind the Weierstrass M-Test, we can now begin to
look at some of its applications. We will first consider an application of the Weierstrass M-Test
in the set of , before moving into applications within the set of .
Example
Show that the real-valued series
is uniformly convergent.
The Weierstrass M-Test gives us the ability to show this without considering any limits.
First, we observe that for any
,
for all k. Then it is easy to see that
. So now let
.
Now we want to show that the series
(our series
) is convergent. This
series converges to by the following Lemma.
LEMMA:
The series
So we now have
converges to
if
.
. Hence,
(our series
). Now by the Weierstrass M-Test we see the series
uniformly convergent on .
is
Roberson 4
Below we see the behavior of the series
the amplitude of the periodic sin function.
, and we can see how
dampens
0.3
0.2
0.1
30
20
10
10
0.1
0.2
0.3
20
30
Roberson 5
We can also look at the series
effects the dampening effect of
graphically and see how this change
.
0.2
0.1
3
2
1
1
0.1
0.2
2
3
Roberson 6
We can also look to see how altering the dampening coefficient may affect the graph. Let the
dampening coefficient be
0.4
0.2
0.5
0.2
0.4
1.0
1.5
2.0
2.5
3.0
Roberson 7
Now to consider an application of the Weierstrass M-Test in the set of
Example
Show that the exponential function
Recall that
.
is uniformly convergent on any bounded set
can be rewritten as the series
.
. Now we will show that this series is
uniformly on some disk D of radius r centered at the origin. To show this we must find some
such that
for all
. Recall that
. Then it follows that
, and that
. We see that
. So let
, so now let
.
We may be able to apply the Weierstrass M-Test if we can show that the series
converges. If we use the Ratio Test we can prove that
is convergent. So now recall:
RATIO TEST:
Given a series
If
If
If
So now we see that
the series
, find
, the series diverges
, the series converges
or the limit fails to exist, then the test is inconclusive.
. Thus by the Ratio Test we see that
converges. Then by the Weierstrass M-Test we see that
convergent on some disk D of radius r centered at the origin.
is uniformly
Roberson 8
References
Ash, R. B., & Novinger, W. P. (n.d.). Complex analysis. Illinois: University of Illinois at UrbanaChampaign, Dept. of Mathematics. Retrieved April 14, 2009, from
http://www.math.uiuc.edu/~r-ash/
Rudin, W. (1976). Principles of mathematical analysis (3rd ed.). Singapore: McGraw-Hill .