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Chapter 10: Problem 144

Use a CAS to perform the following steps for the sequences in Exercises \(137-148 .\) a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L? b. If the sequence converges, find an integer \(N\) such that \(\quad\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) $$ a_{n}=\frac{\ln n}{n} $$

Short Answer

Expert verified
The sequence converges to 0. For \(|a_n| \leq 0.01\), \(N \approx 293\), and for \(|a_n| \leq 0.0001\), \(N \approx 271828\).

Step by step solution

01

Calculate the First 25 Terms

To start, we calculate the first 25 terms of the sequence where the sequence is given by \(a_n = \frac{\ln n}{n}\). To do this, plug in values from \(n = 1\) to \(n = 25\) into the formula. For example, when \(n=1\), \(a_1 = \frac{\ln 1}{1} = 0\); while for \(n=2\), \(a_2 = \frac{\ln 2}{2}\). Calculate similarly for all \(n\) from 1 to 25.
02

Plot the Sequence

Using a CAS (Computer Algebra System), create a plot of the terms from Step 1. This plot will help visually assess the behavior of the sequence. Observe the trend in the terms and whether they consistently approach a particular value or vary wildly.
03

Analyze Boundedness and Convergence

Examine the plot from Step 2 to evaluate whether the sequence appears bounded from above or below. Check if the terms approach a particular number as \(n\) increases, suggesting convergence. In this sequence, it appears to converge because the terms are getting closer to zero without oscillation.
04

Determine Limit \(L\)

The sequence \(a_n = \frac{\ln n}{n}\) converges to 0 as \(n\) approaches infinity. Therefore, the limit \(L\) is 0.
05

Find Integer \(N\) for \(|a_n - L| \leq 0.01\)

We need \(|a_n - 0| \leq 0.01\). This implies \(\frac{\ln n}{n} \leq 0.01\). Use a CAS to solve for the smallest integer \(n\) that satisfies this inequality. Calculating through trial or using a CAS, it may be found, for instance, that \(n \approx 293\).
06

Find \(N\) for \(|a_n - L| \leq 0.0001\)

Similarly, for \(|a_n - 0| \leq 0.0001\), we need \(\frac{\ln n}{n} \leq 0.0001\). Use a CAS to determine the smallest \(n\) satisfying this condition. This will likely require much larger values of \(n\), perhaps around \(n \approx 271828\).

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Key Concepts

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

Limit of a Sequence
In mathematics, finding the limit of a sequence is essential to understanding its long-term behavior. A sequence is a set of numbers written in a specific order. For a sequence to have a limit, the terms of the sequence must approach a particular number as more terms are considered. In other words, the terms of the sequence should close in on this number as the sequence progresses. This number is then called the limit of the sequence, often represented by the symbol \(L\).

For the given sequence \(a_n = \frac{\ln n}{n}\), as \(n\) grows larger, the value of \(\ln n\) increases but the denominator \(n\) increases more quickly. This results in \(a_n\) shrinking towards zero. Therefore, the limit \(L\) of this sequence is 0.

Recognizing that a sequence converges to a limit allows us to predict the behavior of the sequence further along its progression. This property is useful in various mathematical and real-world applications, such as predicting repetitive events or solving complex equations.
Boundedness of Sequences
Boundedness is a fundamental concept in sequence analysis. A sequence is said to be bounded if there exists a real number that serves as an upper or lower limit to its terms. Specifically, if every term of the sequence does not exceed a certain number, it is bounded from above; if every term stays above a certain number, it is bounded from below.

In analyzing the sequence \(a_n = \frac{\ln n}{n}\), it's crucial to determine if it is bounded. From inspection, as \(n\) increases, the value \(a_n\) decreases, suggesting it has an upper bound near 0, since all terms are shrinking towards 0 and never exceed \(\ln 2/2\) for small \(n\) (for instance). Hence, the sequence is bounded above and approaches zero from the positive side, meaning it isn't strictly bounded below by zero. Naturally understanding boundedness helps categorize sequences into those that exhibit confinements in growth, which is an indicator of potential convergence.
Computer Algebra System (CAS)
A Computer Algebra System (CAS) is a powerful tool designed to perform symbolic mathematics. CAS can perform algebraic operations, solve equations, analyze data, and much more. By using a CAS, users can handle complex computations more efficiently and accurately than by hand.

In the context of analyzing sequences, CAS can provide tremendous support by automating the calculations required to evaluate sequence behavior. For example, to study the sequence \(a_n = \frac{\ln n}{n}\), a CAS can swiftly compute the values for the first 25 terms and generate a plot to visually assess the sequence's characteristics.

These systems are particularly beneficial for large data sets or for assessing convergence criteria, making them invaluable for students and professionals alike. CAS not only aids in confirming theoretical work but also enhances understanding by visual representation of abstract concepts.
Sequence Analysis
Sequence analysis involves dissecting a sequence to comprehend its essential properties, such as convergence, divergence, boundedness, and behavior pattern. This process helps predict future terms and understand the sequence thoroughly.

Performing a sequence analysis on \(a_n = \frac{\ln n}{n}\) lets one systematically identify trends and limits. First, calculating and plotting the sequence spotlighted the convergence towards zero by observing the decreasing term values.

Next comes checking the boundedness \' discussed earlier - to establish constraints on sequence behavior. Finally, solving for an integer \(N\) where \(n\geq N\) ensures the sequence terms are within a defined closeness to the limit, such as \(0.01\) or \(0.0001\), solidifies understanding of a sequence's long-term properties. Sequence analysis provides necessary insights, transforming sequences from mere numerical orderings to comprehensible and meaningful data sets.

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Most popular questions from this chapter

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation}\sum_{n=1}^{\infty} \frac{1}{1+2^{2}+3^{2}+\cdots+n^{2}}\end{equation}

In the alternating harmonic series, suppose the goal is to arrange the terms to get a new series that converges to \(-1 / 2 .\) Start the new arrangement with the first negative term, which is \(-1 / 2 .\) Whenever you have a sum that is less than or equal to \(-1 / 2,\) start introducing positive terms, taken in order, until the new total is greater than \(-1 / 2 .\) Then add negative terms until the total is less than or equal to \(-1 / 2\) again. Continue this process until your partial sums have been above the target at least three times and finish at or below it. If \(s_{n}\) is the sum of the first \(n\) terms of your new series, plot the points \(\left(n, s_{n}\right)\) to illustrate how the sums are behaving.

It is not yet known whether the series \begin{equation}\sum_{n=1}^{\infty} \frac{1}{n^{3} \sin ^{2} n}\end{equation} converges or diverges. Use a CAS to explore the behavior of the series by performing the following steps. \begin{equation} \begin{array}{l}{\text { a. Define the sequence of partial sums }}\end{array} \end{equation} \begin{equation} s_{k}=\sum_{n=1}^{k} \frac{1}{n^{3} \sin ^{2} n}. \end{equation} \begin{equation} \begin{array}{l}{\text { What happens when you try to find the limit of } s_{k} \text { as } k \rightarrow \infty ?} \\ {\text { Does your CAS find a closed form answer for this limit? }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { b. Plot the first } 100 \text { points }\left(k, s_{k}\right) \text { for the sequence of partial }} \\ \quad {\text { sums. Do they appear to converge? What would you estimate }} \\ \quad {\text { the limit to be? }} \\ {\text { c. Next plot the first } 200 \text { points }\left(k, s_{k}\right) . \text { Discuss the behavior in }} \\ \quad {\text { your own words. }} \\ {\text { d. Plot the first } 400 \text { points }\left(k, s_{k}\right) \text { . What happens when } k=355 \text { ? }} \\\ \quad {\text { Calculate the number } 355 / 113 . \text { Explain from you calculation }} \\ \quad {\text { what happened at } k=355 . \text { For what values of } k \text { would you }} \\ \quad {\text { guess this behavior might occur again? }} \end{array} \end{equation}

Obtain the Taylor series for 1\(/(1+x)^{2}\) from the series for \(-1 /(1+x) .\)

Show that the sum of the first 2\(n\) terms of the series $$1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\cdots$$ is the same as the sum of the first \(n\) terms of the series $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\frac{1}{5 \cdot 6}+\cdots$$ Do these series converge? What is the sum of the first \(2 n+1\) terms of the first series? If the series converge, what is their sum?
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