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.**
a) \[\left\{ \frac{(2n-1)!}{(2n+1)!} \right\}\]
b) \[a_n = \frac{2^n \sin n}{3^n}\]
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**Explanation:**
The problem poses two sequences for analysis to determine their convergence or divergence. If a sequence converges, the task is to find the limit of that sequence.
### a) \[\left\{ \frac{(2n-1)!}{(2n+1)!} \right\}\]
This sequence involves the factorials of the terms \(2n-1\) and \(2n+1\), which present a more complex factorial relationship that may potentially simplify when factoring terms.
### b) \[a_n = \frac{2^n \sin n}{3^n}\]
This sequence involves an exponential term in the numerator and an exponential term in the denominator, along with a sinusoidal function. The behavior of the sine function and the ratio of the exponential terms will dictate the convergence or divergence of the sequence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22d194fc-eadc-4c3d-a413-33baafd152a9%2F2b98d20b-0c55-4599-82ed-12548d600884%2F0w1hm9c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine whether the sequence converges or diverges. If it converges, find the limit.**
a) \[\left\{ \frac{(2n-1)!}{(2n+1)!} \right\}\]
b) \[a_n = \frac{2^n \sin n}{3^n}\]
---
**Explanation:**
The problem poses two sequences for analysis to determine their convergence or divergence. If a sequence converges, the task is to find the limit of that sequence.
### a) \[\left\{ \frac{(2n-1)!}{(2n+1)!} \right\}\]
This sequence involves the factorials of the terms \(2n-1\) and \(2n+1\), which present a more complex factorial relationship that may potentially simplify when factoring terms.
### b) \[a_n = \frac{2^n \sin n}{3^n}\]
This sequence involves an exponential term in the numerator and an exponential term in the denominator, along with a sinusoidal function. The behavior of the sine function and the ratio of the exponential terms will dictate the convergence or divergence of the sequence.
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