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2. Use limits to determine if the the following sequences converge or diverge. If the sequence converges, determine its limit as an exact value. (a) {an} with an = ln(4n²+1) - ln(n² - 1) (d) {dn} with dn = 3¹/n (b) {bn} with bn = (c) {an} with an = n πn 8n³ + 3 n² 1/3 (e) {bn} with bn = n 2 (¹ + ²/)" (f) {n} with cn = = el-n²

2. Use limits to determine if the the following sequences converge or diverge. If the sequence converges, determine its limit as an exact value. (a) {an} with an = ln(4n²+1) - ln(n² - 1) (d) {dn} with dn = 3¹/n (b) {bn} with bn = (c) {an} with an = n πn 8n³ + 3 n² 1/3 (e) {bn} with bn = n 2 (¹ + ²/)" (f) {n} with cn = = el-n²

Calculus: Early Transcendentals
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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### Sequences and Limits Problem **Problem Statement:** Use limits to determine if the following sequences converge or diverge. If the sequence converges, determine its limit as an exact value. **Sequences:** (a) \( \{a_n\} \) with \( a_n = \ln(4n^2 + 1) - \ln(n^2 - 1) \) (b) \( \{b_n\} \) with \( b_n = \frac{n}{\pi^n} \) (c) \( \{a_n\} \) with \( a_n = \left( 8n^3 + \frac{3}{n^2} \right)^{1/3} \) (d) \( \{d_n\} \) with \( d_n = 3^{1/n} \) (e) \( \{b_n\} \) with \( b_n = \left( 1 + \frac{2}{n} \right)^n \) (f) \( \{c_n\} \) with \( c_n = e^{1-n^2} \) **Instructions:** For each sequence, apply limits to determine convergence or divergence. If convergent, find the exact limit value. Consider using methods like L'Hôpital's Rule, comparison tests, or exponential growth/decay properties as needed.
Transcribed Image Text:### Sequences and Limits Problem **Problem Statement:** Use limits to determine if the following sequences converge or diverge. If the sequence converges, determine its limit as an exact value. **Sequences:** (a) \( \{a_n\} \) with \( a_n = \ln(4n^2 + 1) - \ln(n^2 - 1) \) (b) \( \{b_n\} \) with \( b_n = \frac{n}{\pi^n} \) (c) \( \{a_n\} \) with \( a_n = \left( 8n^3 + \frac{3}{n^2} \right)^{1/3} \) (d) \( \{d_n\} \) with \( d_n = 3^{1/n} \) (e) \( \{b_n\} \) with \( b_n = \left( 1 + \frac{2}{n} \right)^n \) (f) \( \{c_n\} \) with \( c_n = e^{1-n^2} \) **Instructions:** For each sequence, apply limits to determine convergence or divergence. If convergent, find the exact limit value. Consider using methods like L'Hôpital's Rule, comparison tests, or exponential growth/decay properties as needed.
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