Problem 1. Suppose the series (11.1) converges to f(z) for all z suchthat r < z – zol Problem 2. Prove that if the Laurent seriesa„z"represents an even function, thena2k+1 = 0 (k = 0, ±1, ±2,...),while if the series represents an odd function, thenazk = 0 (k = 0, ±1, ±2,...).

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Answered: Problem 2. Prove that if the Laurent…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Problem 4. Does the series n=2 1 √n - 1 converge or diverge? Show all work.
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QUESTION 5 Find the Generating function for the sequence 1,5,25,125,625,. Also find for which value the series converges.
Problem 1. The Alternating Series Test states that if the positive sequence {b„} is (1) decreasing, and (2) convergent to 0, then the series > (-1)"+'bn converges. But what if we drop the assumption that {bn} is decreasing? Is the n=1 result still true? Consider the series defined by 1 1 1 1 1 1 1 1 (-1)"+1bn, - 2 4 3 8. 4 16 2n n=1 where the sequence b, is defined by 1 1 1 1 1 1 1 4' 3'8'4'16’5'32' 1 1 } n 2n (a) Does this sequence {bn} satisfy the assumptions of the Alternating Series Test? Which does it satisfy, and which does it fail? 1 (b) Show that this series diverges. (Hint: This series can also be written as Suppose that it did n 2n n=1 should converge. If you add a certain geometric series to it, you're adding two convergent series together, so you get another convergent series but do you?) (c) Is the Alternating Series Test wrong? Explain why not.
QUESTION 5 Find the Generating function for the sequence 1,5,25,125,625,.. Also find for which value the series converges.
Problem 3: Life is better when there are options. The series can be analyzed in many п(п - 1) n=3 different ways. You will discover them below! (a) Show this series a telescoping series. (You may need to find a function f(x) so that Sp = f(n), where {Sm} is the associated sequence of partial sums, see the hint in the next part.) Find the sum of this telescoping series. (b) Use the Integral Test to determine whether the series converges or diverges. What is the value of the associated improper integral? Be sure to carefully write out your argument. (Hint: write out the telescoping series in the form E3 A - for some constants A, B and apply the integral test to this series.) (c) Does the sum of the series in part (a) match the value of the improper integral from part (b)? Should it? Briefly explain the reasoning that allowed you to make your conclusion. (A picture might be helpful too!) (d) Extra Credit: Use the Limit Comparison Test to determine whether the series converges or diverges.…
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Problem 2. Prove that if the Laurent series represents an even function, then a2k+1 = 0 (k = 0, ±1, ±2,...), %3D while if the series represents an odd function, then