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Use the definition of convergence to prove lim 1 11 7²27-1 = 0; no theorem in

Use the definition of convergence to prove lim 1 11 7²27-1 = 0; no theorem in

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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can you use the same proof like example to do this question?

4. Use the definition of convergence to prove lim¹ = 0; no theorem in Ross 9 is allowed.
Transcribed Image Text:4. Use the definition of convergence to prove lim¹ = 0; no theorem in Ross 9 is allowed.
Formal Proof Let e > 0 and let N max{2, ₁√√54}. Then n > N implies n > hence 54 < €, also 27n3n+24. and hence as desired. 27n hence lim = [4n³ + 3n n³ - 6 3n+ 24 n³ 6 23 n³/2 < €. Since n > 2, we have n ≤n³ - 6 and Thus n > N implies 2 < = = lim 4n³ + 3n n³-6 27n }n3 4 Example 3 illustrates direct proofs of even rather simple limits can get complicated. With the limit theorems of §9 we would just write 54 n² < €₂ 4|<₁₂ 4 < €, = 54 lim 4+3 · lim(2) lim 1-6-lim(3) € 9 = 4.
Transcribed Image Text:Formal Proof Let e > 0 and let N max{2, ₁√√54}. Then n > N implies n > hence 54 < €, also 27n3n+24. and hence as desired. 27n hence lim = [4n³ + 3n n³ - 6 3n+ 24 n³ 6 23 n³/2 < €. Since n > 2, we have n ≤n³ - 6 and Thus n > N implies 2 < = = lim 4n³ + 3n n³-6 27n }n3 4 Example 3 illustrates direct proofs of even rather simple limits can get complicated. With the limit theorems of §9 we would just write 54 n² < €₂ 4|<₁₂ 4 < €, = 54 lim 4+3 · lim(2) lim 1-6-lim(3) € 9 = 4.
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Step 1: Definition of convergence of a real sequence

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