Σ'an where an n=1 = (−1)n+5 ( n² n? + 6 n? +5,
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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Question
1) Consider the series in the image below:
a. Would this series be alternating at some point? (Yes/No)
b. Are the terms in this series nonincreasing in its magnitude? (Yes/No)
c. What would be the limit for this series?
d. Based on the limit, can alternating series test be applied here? (Yes/No)
e. Would this series converge or diverge?
![The expression shown involves a series and is given by:
\[
\sum_{n=1}^{\infty} a_n
\]
where
\[
a_n = (-1)^{n+5} \left( \frac{n^2 + 6}{n^2 + 5} \right).
\]
This series is a representation of an infinite sum where the general term \( a_n \) is defined as a product of a sign-changing factor \( (-1)^{n+5} \) and a rational function \( \left( \frac{n^2 + 6}{n^2 + 5} \right) \). The factor \( (-1)^{n+5} \) alternates the sign of each term in the sequence depending on the value of \( n \). This type of series is known as an alternating series because the terms alternate in sign. The fraction represents a ratio of quadratic polynomials in \( n \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb96d91a2-1904-4455-9ca0-61adba14ab53%2F792e793e-4cda-4cf2-84a0-aedf0ef28c49%2Frcikdah_processed.png&w=3840&q=75)
Transcribed Image Text:The expression shown involves a series and is given by:
\[
\sum_{n=1}^{\infty} a_n
\]
where
\[
a_n = (-1)^{n+5} \left( \frac{n^2 + 6}{n^2 + 5} \right).
\]
This series is a representation of an infinite sum where the general term \( a_n \) is defined as a product of a sign-changing factor \( (-1)^{n+5} \) and a rational function \( \left( \frac{n^2 + 6}{n^2 + 5} \right) \). The factor \( (-1)^{n+5} \) alternates the sign of each term in the sequence depending on the value of \( n \). This type of series is known as an alternating series because the terms alternate in sign. The fraction represents a ratio of quadratic polynomials in \( n \).
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