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Find a sequence of real numbers {a;} such that lim a, does not n-00 xist, but it has 3 subsequential limits of 0,1,2.

Find a sequence of real numbers {a;} such that lim a, does not n-00 xist, but it has 3 subsequential limits of 0,1,2.

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:** Find a sequence of real numbers \(\{a_i\}\) such that \(\lim_{n \to \infty} a_n\) does not exist, but it has 3 subsequential limits of 0, 1, 2. **Explanation:** This mathematical problem requires finding a sequence where the overall limit does not exist, but within the sequence, there are subsequences that specifically converge to 0, 1, and 2. 1. **Sequence Definition:** Identify a sequence pattern where individual subsequences approach distinct limits. 2. **Analysis of Limits:** Check why the limit of the sequence as a whole does not exist, while demonstrating the convergence of subsequences to 0, 1, and 2. By structuring the sequence appropriately, you can exhibit the required characteristics while exploring concepts such as divergence and convergence within different parts of the sequence. This problem involves advanced understanding of limits, subsequences, and convergence in real analysis.
Transcribed Image Text:**Problem Statement:** Find a sequence of real numbers \(\{a_i\}\) such that \(\lim_{n \to \infty} a_n\) does not exist, but it has 3 subsequential limits of 0, 1, 2. **Explanation:** This mathematical problem requires finding a sequence where the overall limit does not exist, but within the sequence, there are subsequences that specifically converge to 0, 1, and 2. 1. **Sequence Definition:** Identify a sequence pattern where individual subsequences approach distinct limits. 2. **Analysis of Limits:** Check why the limit of the sequence as a whole does not exist, while demonstrating the convergence of subsequences to 0, 1, and 2. By structuring the sequence appropriately, you can exhibit the required characteristics while exploring concepts such as divergence and convergence within different parts of the sequence. This problem involves advanced understanding of limits, subsequences, and convergence in real analysis.
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