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Chapter 8: Problem 39

Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. $$a_{n}=1+\cos \frac{1}{n}$$

Short Answer

Expert verified
Answer: The limit of the sequence is \(2\).

Step by step solution

01

Analyze the sequence

In order to find the limit, first, examine the \(a_n\) sequence: \(a_n = 1 + \cos \frac{1}{n}\). The sequence is the sum of two components: a constant and a cosine function with the argument \(\frac{1}{n}\).
02

Apply the limit

To find the limit of the sequence as \(n\) approaches infinity, take the limit of the sequence \(a_n\): $$\lim_{n\to\infty} a_n = \lim_{n\to\infty} \left(1 + \cos \frac{1}{n}\right)$$
03

Separate the limit into two parts

Now, we can separate the limit into two parts: $$\lim_{n\to\infty} a_n = \lim_{n\to\infty} 1 + \lim_{n\to\infty} \cos \frac{1}{n}$$
04

Calculate the limits

First, find the limit of the constant component: $$\lim_{n\to\infty} 1 = 1$$ Now, for the cosine component, as \(n\) approaches infinity, \(\frac{1}{n}\) approaches \(0\). Therefore, we have: $$\lim_{n\to\infty} \cos \frac{1}{n} = \cos \lim_{n\to\infty} \frac{1}{n} = \cos(0)$$ Since \(\cos(0) = 1\), the limit of the cosine component is also \(1\).
05

Combine the limits

Now, we can combine the limits of the two components: $$\lim_{n\to\infty} a_n = 1 + 1 = 2$$
06

Graph the sequence

Lastly, to verify the result with a graphing utility, you can plot the sequence \(a_n = 1 + \cos \frac{1}{n}\). As \(n\) increases, the values of \(a_n\) approach \(2\), confirming the answer. In conclusion, the limit of the sequence \(a_n = 1 + \cos \frac{1}{n}\) as \(n\) approaches infinity is \(2\).

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Key Concepts

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

Sequences
Sequences are an essential part of calculus that describe ordered collections of numbers, often defined by specific rules or formulas. They are like functions, but instead of mapping every real number to another real number, they map positive integers to real numbers. In this exercise, the sequence is given by the formula \(a_n = 1 + \cos \frac{1}{n}\). This means each term in the sequence depends on the integer \(n\).
  • \(n\) is the term number, often called the index.
  • \(a_n\) is the value of the sequence at that particular \(n\).
Understanding how a sequence behaves as \(n\) becomes very large is crucial in calculus. In many cases, we are interested in finding if the sequence approaches a specific value, known as the limit of the sequence. This behavior can give us insight into the properties of the function or pattern described by the sequence.
Limits
The concept of limits is fundamental in calculus. It helps us understand what value a function or sequence approaches as the input approaches a certain point. In this exercise, we are interested in finding the limit of the sequence \(a_n = 1 + \cos \frac{1}{n}\) as \(n\) approaches infinity.
The idea is to determine what value \(a_n\) gets closer to as \(n\) increases without bound. To find this limit, we break the sequence into two parts:
  • The constant part \(1\), which always remains the same as \(n\) changes. The limit of a constant is simply the constant itself, so \(\lim_{n\to\infty} 1 = 1\).
  • The cosine part \(\cos \frac{1}{n}\), which involves an angle that gets closer to 0 as \(n\) becomes very large. Knowing that \(\cos(0) = 1\), we find that \(\lim_{n\to\infty} \cos \frac{1}{n} = 1\).
By combining these separate limits, we find that, as \(n\) approaches infinity, the sequence \(a_n\) approaches \(1 + 1 = 2\).
Understanding limits in sequences and functions provides clarity about long-term behavior and convergence, which is especially useful in real-world applications and mathematical modeling.
Graphing Utilities
Graphing utilities, such as graphing calculators or software, are powerful tools that allow us to visually analyze functions and sequences. They provide a graphical representation, making it easier to understand and confirm mathematical results by observing patterns and behavior at large values of \(n\).
In this exercise, you can use a graphing utility to plot the sequence \(a_n = 1 + \cos \frac{1}{n}\). As you increase \(n\), you should see the values of the sequence moving towards a horizontal line at \(y = 2\). This graphical confirmation supports the analytical result we found for the limit.
Graphing tools often provide additional features:
  • Zooming in and out to check detailed behavior at specific intervals.
  • Animating the graph to see the progression dynamically.
These utilities are extremely useful not only for verifying calculations but also for gaining intuition about how sequences and functions behave as their inputs grow.

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Most popular questions from this chapter

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 3^{-k}$$

Marie takes out a \(\$ 20,000\) loan for a new car. The loan has an annual interest rate of \(6 \%\) or, equivalently, a monthly interest rate of \(0.5 \% .\) Each month, the bank adds interest to the loan balance (the interest is always \(0.5 \%\) of the current balance), and then Marie makes a \(\$ 200\) payment to reduce the loan balance. Let \(B_{n}\) be the loan balance immediately after the \(n\) th payment, where \(B_{0}=\$ 20,000\). a. Write the first five terms of the sequence \(\left\\{B_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{B_{n}\right\\}\). c. Determine how many months are needed to reduce the loan balance to zero.

Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive integer \(N\) and call it \(a_{0} .\) This is the seed of a sequence. The rest of the sequence is generated as follows: For \(n=0,1,2, \ldots\) $$a_{n+1}=\left\\{\begin{array}{ll} a_{n} / 2 & \text { if } a_{n} \text { is even } \\ 3 a_{n}+1 & \text { if } a_{n} \text { is odd .} \end{array}\right.$$ However, if \(a_{n}=1\) for any \(n,\) then the sequence terminates. a. Compute the sequence that results from the seeds \(N=2,3\), \(4, \ldots, 10 .\) You should verify that in all these cases, the sequence eventually terminates. The hailstone conjecture (still unproved) states that for all positive integers \(N\), the sequence terminates after a finite number of terms. b. Now define the hailstone sequence \(\left\\{H_{k}\right\\},\) which is the number of terms needed for the sequence \(\left\\{a_{n}\right\\}\) to terminate starting with a seed of \(k\). Verify that \(H_{2}=1, H_{3}=7\), and \(H_{4}=2\). c. Plot as many terms of the hailstone sequence as is feasible. How did the sequence get its name? Does the conjecture appear to be true?

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. $$a_{n+1}=\frac{1}{2}\left(a_{n}+2 / a_{n}\right) ; a_{0}=2$$

The expression $$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+}}}}.$$ where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+1 / a_{n},\) for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n},\) provided the limit exists. b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of $$a+\frac{b}{a+\frac{b}{a+\frac{b}{a+\frac{b}{a+}}}}$$ where \(a\) and \(b\) are positive real numbers.
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