Problem X.4Suppose the radius of convergence of the power seriesCn" is R. What is the radius of conver-n=0gence of the power series Cna"?

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Answered: Problem X.4 Suppose the radius of…

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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12. Determine if the series is absolutely convergent, conditionally convergent, or divergent using one of the following (divergence test, integral test, comparison test, limit comparison test, alternating series test, ratio test, root test).
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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 X.4 Suppose the radius of convergence of the power series CnI" is R. What is the radius of conver- n=0 gence of the power series Chang n=0