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Chapter 9: Problem 42

Find the limit (if possible) of the sequence. $$ a_{n}=\cos \frac{2}{n} $$

Short Answer

Expert verified
The limit of the sequence \(a_n = \cos \frac{2}{n}\) as \(n\) approaches infinity is 1.

Step by step solution

01

Understand the behavior of function

To find the limit of a sequence, it can often be helpful to first understand the behavior of the function being used in the sequence. In our case, this is the cosine function. As \(n\) increases, \( \frac{2}{n} \) decreases and approaches 0. The cosine of 0 is 1. Therefore, as \(n\) approaches infinity, \( \cos \frac{2}{n} \) approaches \(\cos(0) = 1\).
02

Apply the limit

Next, formally apply the limit. We want to find the value that the sequence approaches as \(n\) goes to infinity. We denote this as \( \lim_{{n \to \infty}} \cos \frac{2}{n} \). Considering our analysis in step 1, we can say that as \(n\) increases indefinitely \( \frac{2}{n} \) will approach 0. Therefore, \( \cos \frac{2}{n} \) will approach \(\cos(0)\).
03

Evaluate the limit

Finally, evaluate the limit. We know that \( \cos(0) = 1 \). Therefore, \( \lim_{{n \to \infty}} \cos \frac{2}{n} = 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function is one of the fundamental components of trigonometry, harboring significant relevance in various areas of mathematics, including sequence limits. It is defined as the ratio of the adjacent side to the hypotenuse of a right-angled triangle. But more so, the cosine function, denoted by \( \cos(x) \), is a periodic and even function, meaning it repeats its values in regular intervals and is symmetric about the y-axis. As a result, the cosine function is predictable and continuous, making it easier to interpret in terms of limits.

For small angles, the cosine function is approximated as \( \cos(x) \approx 1 - \frac{x^2}{2} \) when \( x \) is close to zero. This relationship is crucial when evaluating limits involving the cosine function as \( x \) approaches zero. Understanding the behavior of the \(\frac{2}{n} \) portion as \( n \) increases is essential for determining the limit of the sequence involving \( \cos \). When this argument of the cosine function approaches zero, the value of the cosine function itself approaches 1. This piece of knowledge is pivotal for grasping the fundamental behavior of the cosine function in sequence limits.
Convergence of Sequences
Convergence is a key concept while dealing with sequences in mathematics. A sequence is said to converge if its terms approach a specific value, known as the limit, as the number of terms goes to infinity. Speaking of the sequence \( a_n \), we are looking at an infinite list of numbers derived by applying a formula, here involving the cosine function, to every natural number \( n \).

The question whether this particular sequence converges hinges on the behavior of its terms as \( n \) gets arbitrarily large. In the example of \( a_{n}=\cos \frac{2}{n} \), the sequence converges because as \( n \) grows, \( \frac{2}{n} \) gets smaller and smaller, approaching zero; hence, the \( \cos \) part of the terms approaches \( \cos(0) \) which is a definitive value, specifically 1. It indicates that no matter how far out into the sequence you go, the terms will get closer and closer to 1, thus illustrating the convergence of this sequence to the limit of 1. Understanding whether and how a sequence converges is instrumental in analyzing its long-term behavior and finding its limit.
Evaluating Limits
Evaluating limits is a cornerstone of calculus, involving finding the value that a sequence or function

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Most popular questions from this chapter

(a) You delete a finite number of terms from a divergent series. Will the new series still diverge? Explain your reasoning. (b) You add a finite number of terms to a convergent series. Will the new series still converge? Explain your reasoning.

Let \(f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}\) (a) Find the interval of convergence of \(f\). (b) Show that \(f^{\prime}(x)=f(x)\). (c) Show that \(f(0)=1\). (d) Identify the function \(f\).

The random variable \(n\) represents the number of units of a product sold per day in a store. The probability distribution of \(n\) is given by \(P(n) .\) Find the probability that two units are sold in a given day \([P(2)]\) and show that \(P(1)+P(2)+P(3)+\cdots=1\) $$ P(n)=\frac{1}{3}\left(\frac{2}{3}\right)^{n} $$

show that the function represented by the power series is a solution of the differential equation. $$ y=1+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{4 n}}{2^{2 n} n ! \cdot 3 \cdot 7 \cdot 11 \cdot \cdots(4 n-1)}, y^{\prime \prime}+x^{2} y=0 $$

Bessel Function The Bessel function of order 0 is \(J_{0}(x)=\sum_{k=0}^{\infty} \frac{(-1)^{k} x^{2 k}}{2^{2 k}(k !)^{2}}\) (a) Show that the series converges for all \(x\). (b) Show that the series is a solution of the differential equation \(x^{2} J_{0}{ }^{n}+x J_{0}{ }^{\prime}+x^{2} J_{0}=0\) (c) Use a graphing utility to graph the polynomial composed of the first four terms of \(J_{0}\). (d) Approximate \(\int_{0}^{1} J_{0} d x\) accurate to two decimal places.
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