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7. Suppose lim nan = A +0 where an is not necessarily nonnegative. Prove >a,n diverges. n=1

7. Suppose lim nan = A +0 where an is not necessarily nonnegative. Prove >a,n diverges. n=1

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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Certainly! Here is a transcription suitable for an educational website: --- **7. Mathematical Problem:** Suppose \(\lim_{{n \to \infty}} n a_n = A \neq 0\) where \(a_n\) is not necessarily nonnegative. Prove that the series \[ \sum_{{n=1}}^{\infty} a_n \] diverges. --- In this problem, you are given a sequence \(a_n\) with the condition that the limit of the product of this sequence and \(n\) approaches a non-zero constant \(A\) as \(n\) approaches infinity. The task is to demonstrate that the infinite series formed by summing the terms \(a_n\) diverges.
Transcribed Image Text:Certainly! Here is a transcription suitable for an educational website: --- **7. Mathematical Problem:** Suppose \(\lim_{{n \to \infty}} n a_n = A \neq 0\) where \(a_n\) is not necessarily nonnegative. Prove that the series \[ \sum_{{n=1}}^{\infty} a_n \] diverges. --- In this problem, you are given a sequence \(a_n\) with the condition that the limit of the product of this sequence and \(n\) approaches a non-zero constant \(A\) as \(n\) approaches infinity. The task is to demonstrate that the infinite series formed by summing the terms \(a_n\) diverges.
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