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
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:
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**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae372ae2-2feb-4e9a-b164-90567cadbb97%2Ffb80a51c-1339-4cdc-b699-559d06f127ba%2Fb8gnt0r_processed.png&w=3840&q=75)
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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