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is just a convenient notational shorthand telling us that the sequence diverges by becoming arbitrarily large. Problem 76. Suppose lim a, = o and lim bn = -o and a €R. Prove or give a counterezample: CONVERGENCE OF SEQUENCES AND SERIES 86 (a) lim a, + b, = 00

is just a convenient notational shorthand telling us that the sequence diverges by becoming arbitrarily large. Problem 76. Suppose lim a, = o and lim bn = -o and a €R. Prove or give a counterezample: CONVERGENCE OF SEQUENCES AND SERIES 86 (a) lim a, + b, = 00

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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8:32 ll 1 Safari < RealAnalysis-ISBN-fix... is just a convenient notational shorthand telling us that the sequence diverges by becoming arbitrarily large. Problem 76. Suppose lim an = 0 and lim b, = -0 and a e R. Prove or give a counterezample: CONVERGENCE OF SEQUENCES AND SERIES 86 (a) lim a, + b, = 00 (b) lim a,b, = -00 (c) lim aa, = 00 (d) lim ab, = -00 Finally, a sequence can diverge in other ways as the following problem dis- plays. Problem 77. Show that each of the following sequences diverge. (a) ((-1)") (b) (-1)" п)я-1 (1 if n = 2P for some peN (e) an = ! otherwise, Problem 78. Suppose that (a,) diverges but not to infinity and that a is a real number. What conditions on a will guarantee that: (a) (aa,) converges? (b) (aa,) diverges? Problem 79. Show that if r| > 1 then (r") diverges. Will it diverge to infinity? Additional Problems Problem 80. Prove that if lim,- Sn = 8 then lim- |Sn| = |s|. Prove that the converse is true uhen s = 0, but it is not necessarily true otherwise. Problem 81. (a) Let (sn) and (tn) be sequences with s, < tn, Vn. Suppose lim,- 8n = s and lim,- t, = t. Prove s < t. [Hint: Assume for contradiction, that 8>t and use the definition of converaence with e = to produce an n Next 1 Dashboard Calendar To Do Notifications Inbox 因
Transcribed Image Text:8:32 ll 1 Safari < RealAnalysis-ISBN-fix... is just a convenient notational shorthand telling us that the sequence diverges by becoming arbitrarily large. Problem 76. Suppose lim an = 0 and lim b, = -0 and a e R. Prove or give a counterezample: CONVERGENCE OF SEQUENCES AND SERIES 86 (a) lim a, + b, = 00 (b) lim a,b, = -00 (c) lim aa, = 00 (d) lim ab, = -00 Finally, a sequence can diverge in other ways as the following problem dis- plays. Problem 77. Show that each of the following sequences diverge. (a) ((-1)") (b) (-1)" п)я-1 (1 if n = 2P for some peN (e) an = ! otherwise, Problem 78. Suppose that (a,) diverges but not to infinity and that a is a real number. What conditions on a will guarantee that: (a) (aa,) converges? (b) (aa,) diverges? Problem 79. Show that if r| > 1 then (r") diverges. Will it diverge to infinity? Additional Problems Problem 80. Prove that if lim,- Sn = 8 then lim- |Sn| = |s|. Prove that the converse is true uhen s = 0, but it is not necessarily true otherwise. Problem 81. (a) Let (sn) and (tn) be sequences with s, < tn, Vn. Suppose lim,- 8n = s and lim,- t, = t. Prove s < t. [Hint: Assume for contradiction, that 8>t and use the definition of converaence with e = to produce an n Next 1 Dashboard Calendar To Do Notifications Inbox 因
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