Problem 8 (Proof of the Root Test). For this problem, you will prove the Root Test theorem. You may not use the Root Test itself at any point during this problem. (You may want to follow along with the proof of the Ratio Test in Rogowski, p. 576.) Let > an be a positive series, and assume L = lim van exists. n=0 (a) Suppose L < 1. Show that this series converges. (Hint: Choose R with L < R < 1, and explain why an < R" for all n sufficiently large, say n> M for some M. Then compare with the geometric series > R".) n=M (b) Show that the series diverges if L> 1.

Problem 8 (Proof of the Root Test). For this problem, you will prove the Root Test theorem. You may not use the Root Test itself at any point during this problem. (You may want to follow along with the proof of the Ratio Test in Rogowski, p. 576.) Let > an be a positive series, and assume L = lim van exists. n=0 (a) Suppose L < 1. Show that this series converges. (Hint: Choose R with L < R < 1, and explain why an < R" for all n sufficiently large, say n> M for some M. Then compare with the geometric series > R".) n=M (b) Show that the series diverges if L> 1.

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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Question

Problem 8 (Proof of the Root Test). For this problem, you will prove the Root Test theorem. You may not
use the Root Test itself at any point during this problem. (You may want to follow along with the proof of the Ratio
Test in Rogowski, p. 576.)
Let > an be a positive series, and assume L = lim v/an exists.
n→∞
n=0
(a) Suppose L < 1. Show that this series converges. (Hint: Choose R with L < R< 1, and explain why an < R"
for all n sufficiently large, say n > M for some M. Then compare with the geometric series > R".)
n=M
(b) Show that the series diverges if L> 1.

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(2) An interesting fact is that the Root Test (stated at the end of the previous problem) is stronger than the Ratio Test. Proving this fact would require showing two things: First, one can show that any time the Ratio Test limit exists and equals some L then the Root Test limit also exists and equals the same L. Second, one can find examples of series where the Root Test works but the Ratio Test does not. Your task in this problem is to do the second thing above. To be more specific, find an example of a series a, such that • lim Vlan[ exists and equals a number less than 1, meaning the series converges absolutely by the Root Test, and • lim does not exist. Give a brief explanation why your example satisfics the above two properties.
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