Problem 2. Prove that if the Laurent seriesE anz"represents an even function, thena2k+1 = 0 (k = 0, ±1, ±2,...),while if the series represents an odd function, thena2k = 0 (k = 0, ±1, ±2,...).

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Answered: Problem 2. Prove that if the Laurent…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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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.
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Problem 2. Prove that if the Laurent series represents an even function, then a2k+1 = 0 (k = 0, ±1, ±2,...), while if the series represents an odd function, then a2k = 0 (k = 0, ±1, ±2,...). %3D %3D