Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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(message for expert: please don't answer with typed how to find please last time i got one like that i could barely figure out where the answer was and also whoever does it like that doesn't make it easy to tell what is a fraction or super/subscript)
![**Problem Statement**
Determine the convergence or divergence of the sequence with the given \( n \)th term. If the sequence converges, find its limit.
**Sequence Definition**
Given sequence:
\[ a_n = \frac{n}{n^6 + 6} \]
**Analysis**
To determine the convergence or divergence of the sequence, we will analyze the behavior of the nth term as \( n \) approaches infinity. If the sequence converges, we will calculate its limit.
Consider the expression:
\[ a_n = \frac{n}{n^6 + 6} \]
As \( n \) becomes very large, the dominant term in the denominator \( n^6 + 6 \) is \( n^6 \). Thus, the expression can be simplified for large \( n \) as follows:
\[ a_n \approx \frac{n}{n^6} = \frac{1}{n^5} \]
As \( n \to \infty \), \(\frac{1}{n^5} \to 0 \).
Therefore, the sequence converges and its limit is 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F45235a53-69d6-46fa-83a8-de0bf8bf66d1%2Fd002b3df-b4ac-4bc3-a650-ba2cfc092bd8%2F2uee07_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Determine the convergence or divergence of the sequence with the given \( n \)th term. If the sequence converges, find its limit.
**Sequence Definition**
Given sequence:
\[ a_n = \frac{n}{n^6 + 6} \]
**Analysis**
To determine the convergence or divergence of the sequence, we will analyze the behavior of the nth term as \( n \) approaches infinity. If the sequence converges, we will calculate its limit.
Consider the expression:
\[ a_n = \frac{n}{n^6 + 6} \]
As \( n \) becomes very large, the dominant term in the denominator \( n^6 + 6 \) is \( n^6 \). Thus, the expression can be simplified for large \( n \) as follows:
\[ a_n \approx \frac{n}{n^6} = \frac{1}{n^5} \]
As \( n \to \infty \), \(\frac{1}{n^5} \to 0 \).
Therefore, the sequence converges and its limit is 0.
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