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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Transcribed Image Text:**Problem Statement:**
Find the limit of the sequence \( a_n = \frac{\cos n}{4^n} \).
**Solution:**
To determine the limit of the sequence as \( n \) approaches infinity, observe the following:
1. The function \(\cos n\) oscillates between -1 and 1 for all integer values of \(n\).
2. The denominator \(4^n\) grows exponentially as \(n\) increases.
3. Since the numerator \(\cos n\) remains bounded while the denominator increases without bound, the terms of the sequence \(a_n\) approach zero.
Therefore, the limit of the sequence \( a_n = \frac{\cos n}{4^n} \) as \( n \rightarrow \infty \) is 0.
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