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### ObjectiveOur 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: \[ = 0 < 1 \]### ConclusionThe process shows that \( \left| \frac{a_{n+1}}{a_n} \right| \) approaches 0 as \( n \) approaches infinity, confirming the convergence

They did not cancel out . We only put the value 1 in both function

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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