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4. Prove: if an/bn (Here, as always, L represents a finite number, not oo.) L, ba #0 for any n, and bn 0, then an → 0.

4. Prove: if an/bn (Here, as always, L represents a finite number, not oo.) L, ba #0 for any n, and bn 0, then an → 0.

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 Statement:** 4. Prove: if \( \frac{a_n}{b_n} \rightarrow L \), \( b_n \neq 0 \) for any \( n \), and \( b_n \rightarrow 0 \), then \( a_n \rightarrow 0 \). (Here, as always, \( L \) represents a finite number, not \( \infty \).) --- *Explanation:* This problem involves sequences \( a_n \) and \( b_n \). The given conditions are: - The quotient \( \frac{a_n}{b_n} \) approaches a finite limit \( L \). - The sequence \( b_n \) is non-zero for all \( n \) and approaches zero. The goal is to prove that under these conditions, the sequence \( a_n \) also approaches zero.
Transcribed Image Text:**Problem Statement:** 4. Prove: if \( \frac{a_n}{b_n} \rightarrow L \), \( b_n \neq 0 \) for any \( n \), and \( b_n \rightarrow 0 \), then \( a_n \rightarrow 0 \). (Here, as always, \( L \) represents a finite number, not \( \infty \).) --- *Explanation:* This problem involves sequences \( a_n \) and \( b_n \). The given conditions are: - The quotient \( \frac{a_n}{b_n} \) approaches a finite limit \( L \). - The sequence \( b_n \) is non-zero for all \( n \) and approaches zero. The goal is to prove that under these conditions, the sequence \( a_n \) also approaches zero.
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