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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 .