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!
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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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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