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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 satisfies the above two properties.

(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 satisfies the above two properties.

Advanced Engineering Mathematics
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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(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.
Transcribed Image Text:(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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