Determine whether the sequence converges or diverges. If it converges, find the limit. (If the sequence diverges, enter DIVERGES.) an = n+8 49n + 8 lim a= DIVERGES n→∞ X
Determine whether the sequence converges or diverges. If it converges, find the limit. (If the sequence diverges, enter DIVERGES.) an = n+8 49n + 8 lim a= DIVERGES n→∞ X
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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![**Determine whether the sequence converges or diverges. If it converges, find the limit. (If the sequence diverges, enter DIVERGES.)**
The given sequence is defined as:
\[ a_n = \sqrt{\frac{n + 8}{49n + 8}} \]
To find the limit, we need to evaluate:
\[ \lim_{n \to \infty} a_n \]
The image shows the result:
\[ \lim_{n \to \infty} a_n = \text{DIVERGES} \]
An "X" mark is included next to "DIVERGES" indicating a wrong answer.
### Analysis:
To determine the correct limit, we need to first simplify the expression inside the square root:
\[ \frac{n + 8}{49n + 8} \]
For large values of \( n \), the dominant terms in the numerator and denominator are \( n \) and \( 49n \), respectively. Hence, we can approximate the fraction as:
\[ \frac{n (1 + \frac{8}{n})}{49n (1 + \frac{8}{49n})} \approx \frac{1 + \frac{8}{n}}{49 (1 + \frac{8}{49n})} \approx \frac{1}{49} \]
Thus:
\[ a_n = \sqrt{\frac{1}{49}} = \frac{1}{7} \]
So, the correct limit is:
\[ \lim_{n \to \infty} a_n = \frac{1}{7} \]
It looks like the initial assessment of divergence was incorrect based on the detailed analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbe02b51-f32f-4987-98d3-1e9fc879ae68%2F2f377ad4-99c5-416c-af71-23a9f6b3eab2%2F4v6nokh_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine whether the sequence converges or diverges. If it converges, find the limit. (If the sequence diverges, enter DIVERGES.)**
The given sequence is defined as:
\[ a_n = \sqrt{\frac{n + 8}{49n + 8}} \]
To find the limit, we need to evaluate:
\[ \lim_{n \to \infty} a_n \]
The image shows the result:
\[ \lim_{n \to \infty} a_n = \text{DIVERGES} \]
An "X" mark is included next to "DIVERGES" indicating a wrong answer.
### Analysis:
To determine the correct limit, we need to first simplify the expression inside the square root:
\[ \frac{n + 8}{49n + 8} \]
For large values of \( n \), the dominant terms in the numerator and denominator are \( n \) and \( 49n \), respectively. Hence, we can approximate the fraction as:
\[ \frac{n (1 + \frac{8}{n})}{49n (1 + \frac{8}{49n})} \approx \frac{1 + \frac{8}{n}}{49 (1 + \frac{8}{49n})} \approx \frac{1}{49} \]
Thus:
\[ a_n = \sqrt{\frac{1}{49}} = \frac{1}{7} \]
So, the correct limit is:
\[ \lim_{n \to \infty} a_n = \frac{1}{7} \]
It looks like the initial assessment of divergence was incorrect based on the detailed analysis.
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