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Consider the with $2m 1 1 1 1 1 1 2' 2' 4' 4' 8' 8' sequence 1,1, S2m+1 = 2-m. (Sn) n=0 11/n Sn Compute |sn|¹/" and Sn+¹ when n is even and then when it is odd. Thus find all expressions in Theorem 12.3 and hence conclude that the converse of (+) is false.

Consider the with $2m 1 1 1 1 1 1 2' 2' 4' 4' 8' 8' sequence 1,1, S2m+1 = 2-m. (Sn) n=0 11/n Sn Compute |sn|¹/" and Sn+¹ when n is even and then when it is odd. Thus find all expressions in Theorem 12.3 and hence conclude that the converse of (+) is false.

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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Analysis

Theorem 12.3. Let (sn) be a non-zero sequence. Then lim inf Sn+1 Sn In particular, lim < lim inf|s₁|¹/¹ ≤ lim sup |sn|¹/¹ ≤ lim sup Sn+1 Sn = L ⇒ lim|sn|¹/n = L Sn+1 Sn (+)
Transcribed Image Text:Theorem 12.3. Let (sn) be a non-zero sequence. Then lim inf Sn+1 Sn In particular, lim < lim inf|s₁|¹/¹ ≤ lim sup |sn|¹/¹ ≤ lim sup Sn+1 Sn = L ⇒ lim|sn|¹/n = L Sn+1 Sn (+)
Consider the sequence with $2m (Sn) n=0 = 1,1, = S2m+1 1 1 1 1 1 1 2'2' 4' 4'8'8 = 2-m. Sn Sn Compute |sn|¹/n and n+1 when n is even and then when it is odd. Thus find all expressions in Theorem 12.3 and hence conclude that the converse of (+) is false.
Transcribed Image Text:Consider the sequence with $2m (Sn) n=0 = 1,1, = S2m+1 1 1 1 1 1 1 2'2' 4' 4'8'8 = 2-m. Sn Sn Compute |sn|¹/n and n+1 when n is even and then when it is odd. Thus find all expressions in Theorem 12.3 and hence conclude that the converse of (+) is false.
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