Find the limit of the sequence as n approaches infinity: In 1 an 2n + 2
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
![**Problem Statement:**
Find the limit of the sequence as \( n \) approaches infinity:
\[
a_n = \frac{1n - 1}{2n + 2}
\]
---
**Detailed Explanation:**
The sequence given is:
\[
a_n = \frac{n - 1}{2n + 2}
\]
To find the limit as \( n \) approaches infinity, simplify the expression:
1. Divide both the numerator and the denominator by \( n \):
\[
a_n = \frac{\frac{n}{n} - \frac{1}{n}}{\frac{2n}{n} + \frac{2}{n}} = \frac{1 - \frac{1}{n}}{2 + \frac{2}{n}}
\]
2. As \( n \) approaches infinity, the terms \(\frac{1}{n}\) and \(\frac{2}{n}\) approach zero. This simplifies the expression to:
\[
a_n = \frac{1 - 0}{2 + 0} = \frac{1}{2}
\]
**Conclusion:**
The limit of the sequence as \( n \) approaches infinity is \(\frac{1}{2}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F296ff3eb-b3c1-450b-9ede-4efaa8667733%2F658634c1-dd01-4259-89cf-3c71a1ce1987%2Fghmc8zr_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the limit of the sequence as \( n \) approaches infinity:
\[
a_n = \frac{1n - 1}{2n + 2}
\]
---
**Detailed Explanation:**
The sequence given is:
\[
a_n = \frac{n - 1}{2n + 2}
\]
To find the limit as \( n \) approaches infinity, simplify the expression:
1. Divide both the numerator and the denominator by \( n \):
\[
a_n = \frac{\frac{n}{n} - \frac{1}{n}}{\frac{2n}{n} + \frac{2}{n}} = \frac{1 - \frac{1}{n}}{2 + \frac{2}{n}}
\]
2. As \( n \) approaches infinity, the terms \(\frac{1}{n}\) and \(\frac{2}{n}\) approach zero. This simplifies the expression to:
\[
a_n = \frac{1 - 0}{2 + 0} = \frac{1}{2}
\]
**Conclusion:**
The limit of the sequence as \( n \) approaches infinity is \(\frac{1}{2}\).
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