Obtain the limit of|| (n+l) cos lim = lim (n+1)! Cos 学) (e+l)e n! cos( = lim n- (n+1)! cos() n!(1) (n+1)m!(1) - kos (#)| = 1] (n+1)x = lim = lim (n+1) Apply the limit, lim || 0+1 = 0 < 1 = to search O Bi 56°F Sunny ^ 3 ô & o
Obtain the limit of|| (n+l) cos lim = lim (n+1)! Cos 学) (e+l)e n! cos( = lim n- (n+1)! cos() n!(1) (n+1)m!(1) - kos (#)| = 1] (n+1)x = lim = lim (n+1) Apply the limit, lim || 0+1 = 0 < 1 = to search O Bi 56°F Sunny ^ 3 ô & o
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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I am not sure why cos((n+1)pi/3) is equal to cos((n)pi/3). I am also unsure as to how they cancel out to then result in lim n->infinity (1/n+1).
![## Understanding the Limit of Sequences
### Objective
Our goal is to obtain and apply the limit of the sequence described by \( \left| \frac{a_{n+1}}{a_n} \right| \).
### Step-by-Step Solution
**Step 1: Obtain the Limit**
- We start by considering the limit:
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]
- Rewrite the expression:
\[
\lim_{n \to \infty} \left| \frac{\frac{(n+1)!}{3}}{\frac{n!}{3}} \right|
\]
- Simplify:
\[
= \lim_{n \to \infty} \left| \frac{n! \cdot \cos\left(\frac{(n+1)\pi}{3}\right)}{(n+1)! \cdot \cos\left(\frac{n\pi}{3}\right)} \right|
\]
- Further simplification:
\[
= \lim_{n \to \infty} \left| \frac{n! \cdot \cos\left(\frac{(n+1)\pi}{3}\right)}{(n+1) \cdot n! \cdot \cos\left(\frac{n\pi}{3}\right)} \right|
\]
- The limit then becomes:
\[
= \lim_{n \to \infty} \frac{1}{n+1}
\]
**Step 2: Apply the Limit**
- Applying the limit results in:
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{\infty + 1}
\]
- Which simplifies to:
\[
= \frac{1}{\infty}
\]
- Resulting in:
\[
= 0 < 1
\]
### Conclusion
The process shows that \( \left| \frac{a_{n+1}}{a_n} \right| \) approaches 0 as \( n \) approaches infinity, confirming the convergence](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2f288b4-9abe-45b3-924a-5abb903bf487%2Ff5517c21-2f91-4e6d-90d0-87031c60d98d%2Fm6cdni_processed.png&w=3840&q=75)
Transcribed Image Text:## Understanding the Limit of Sequences
### Objective
Our goal is to obtain and apply the limit of the sequence described by \( \left| \frac{a_{n+1}}{a_n} \right| \).
### Step-by-Step Solution
**Step 1: Obtain the Limit**
- We start by considering the limit:
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]
- Rewrite the expression:
\[
\lim_{n \to \infty} \left| \frac{\frac{(n+1)!}{3}}{\frac{n!}{3}} \right|
\]
- Simplify:
\[
= \lim_{n \to \infty} \left| \frac{n! \cdot \cos\left(\frac{(n+1)\pi}{3}\right)}{(n+1)! \cdot \cos\left(\frac{n\pi}{3}\right)} \right|
\]
- Further simplification:
\[
= \lim_{n \to \infty} \left| \frac{n! \cdot \cos\left(\frac{(n+1)\pi}{3}\right)}{(n+1) \cdot n! \cdot \cos\left(\frac{n\pi}{3}\right)} \right|
\]
- The limit then becomes:
\[
= \lim_{n \to \infty} \frac{1}{n+1}
\]
**Step 2: Apply the Limit**
- Applying the limit results in:
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{\infty + 1}
\]
- Which simplifies to:
\[
= \frac{1}{\infty}
\]
- Resulting in:
\[
= 0 < 1
\]
### Conclusion
The process shows that \( \left| \frac{a_{n+1}}{a_n} \right| \) approaches 0 as \( n \) approaches infinity, confirming the convergence
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