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Problem 7.4.8. Suppose (sn) is a sequence of positive numbers such that Sn+1 lim = L. Sn (a) Prove that if L < 1, then limn0 Sn = 0. Hint. Choose R with L < R< 1. By the previous problem, 3 N such that if n > N, then Sn+l < R. Let no > N be Sn fixed and show sno+k < R*sn. Conclude that lim00 Sno+k no' O and let n = no + k. (b) Let c be a positive real number. Prove cn lim = 0. n!

Problem 7.4.8. Suppose (sn) is a sequence of positive numbers such that Sn+1 lim = L. Sn (a) Prove that if L < 1, then limn0 Sn = 0. Hint. Choose R with L < R< 1. By the previous problem, 3 N such that if n > N, then Sn+l < R. Let no > N be Sn fixed and show sno+k < R*sn. Conclude that lim00 Sno+k no' O and let n = no + k. (b) Let c be a positive real number. Prove cn lim = 0. n!

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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Problem 7.4.8. Suppose (sn) is a sequence of positive numbers such that (*)- Sn+1 lim = L. Sn (a) Prove that if L < 1, then limn00 Sn = 0. Hint. Choose R with L < R < 1. By the previous problem, 3N such that if n > N, then Sn+1 < R. Let no > N be Sn fixed and show Sno+k < R*s ng. Conclude that lim-0 s no+k = 0 and let n = no + k. (b) Let c be a positive real number. Prove cn lim 0. %3D п!
Transcribed Image Text:Problem 7.4.8. Suppose (sn) is a sequence of positive numbers such that (*)- Sn+1 lim = L. Sn (a) Prove that if L < 1, then limn00 Sn = 0. Hint. Choose R with L < R < 1. By the previous problem, 3N such that if n > N, then Sn+1 < R. Let no > N be Sn fixed and show Sno+k < R*s ng. Conclude that lim-0 s no+k = 0 and let n = no + k. (b) Let c be a positive real number. Prove cn lim 0. %3D п!
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