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Chapter 10: Problem 141

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 \(L ?\) $$ a_{n}=\sin n $$

Short Answer

Expert verified
The sequence \( a_n = \sin n \) diverges and does not converge to a limit.

Step by step solution

01

Calculation and Plotting

Use a Computer Algebra System (CAS) to calculate the first 25 terms of the sequence given by \( a_n = \sin n \). Then, plot these terms on a graph to visually assess their behavior.
02

Analyzing Boundedness and Convergence

Examine the plot and the calculated terms to determine if the sequence is bounded from above or below. Check if the sequence starts to settle towards a fixed value (converging) or if it continues to oscillate indefinitely (diverging). For \( a_n = \sin n \), since \( \sin n \) is periodic and not approaching a specific value as \( n \to \infty \), the sequence diverges.
03

Convergence Limitation Check

Since the sequence doesn't converge (as noted in Step 2), the limit \( L \) does not exist. Thus, identifying an \( N \) such that \( |a_n - L| \leq 0.01 \) is not applicable. Similarly, finding how far in the sequence we need to go to get terms within 0.0001 of \( L \) is also not applicable.

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Key Concepts

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

Boundedness
In mathematics, "boundedness" refers to whether the values of a sequence stay within a fixed set of numbers or bounds. When a sequence is bounded above, it means that its terms never exceed a certain maximum value. Conversely, if a sequence is bounded below, it means none of its terms fall below a certain minimum value.
For the sequence \( a_n = \sin n \), the boundedness is straightforward due to the properties of the sine function. Sine values always lie between -1 and 1. Thus:
  • The sequence is both bounded above by 1 and bounded below by -1.
  • Every term of the sequence will always remain within this range, no matter how large \( n \) becomes.
This inherent limitation, however, does not necessarily imply convergence, which is another critical aspect to consider for sequences.
Convergence and Divergence
Convergence in sequences refers to the behavior where the sequence terms approach a particular number, known as the limit, as \( n \) goes to infinity. A sequence that does not settle on a single value but continues in a dispersed pattern is considered divergent.
For the sequence \( a_n = \sin n \):
  • Since \( \sin n \) is periodic with no pattern of the terms approaching a fixed single value, the sequence diverges.
  • Even though it is bounded, the oscillating nature of sine means it does not converge to a single limit \( L \).
Therefore, tasks such as finding an \( N \) where \( |a_n - L| \leq 0.01 \) do not apply here. It's essential to understand that boundedness does not infer convergence, as demonstrated with this oscillating sequence where boundedness exists, yet convergence does not.
Computer Algebra System (CAS)
A Computer Algebra System (CAS) is a software tool that facilitates symbolic mathematics. It can perform complex algebraic operations like differentiation, integration, and solving equations, making it indispensable for students and professionals dealing with numerical sequences and series.
In the context of analyzing sequences like \( a_n = \sin n \):
  • The CAS can efficiently compute numerous terms of the sequence and generate plots to visualize their behavior.
  • This graphical representation helps users understand whether the sequence is bounded and observe any indications of convergence or divergence.
Using CAS allows easy handling of computations and is especially valuable in examples where manual calculation of numerous sequence terms would be cumbersome. By inputting just a sequence formula, CAS can deliver insights that are difficult to achieve otherwise without advanced computational aid.

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Most popular questions from this chapter

Use series to approximate the values of the integrals with an error of magnitude less than \(10^{-8}\) . \begin{equation} \int_{0}^{0.1} \sqrt{1+x^{4}} d x \end{equation}

Computer Explorations Taylor's formula with \(n=1\) and \(a=0\) gives the linearization of a function at \(x=0 .\) With \(n=2\) and \(n=3\) we obtain the standard quadratic and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions: \begin{equation} \begin{array}{l}{\text { a. For what values of } x \text { can the function be replaced by each }} \\ {\text { approximation with an error less than } 10^{-2} \text { ? }} \\ {\text { b. What is the maximum error we could expect if we replace the }} \\ {\text { function by each approximation over the specified interval? }}\end{array} \end{equation} Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises \(53-58 .\) \begin{equation} \begin{array}{l}{\text { Step } 1 : \text { Plot the function over the specified interval. }} \\ {\text { Step } 2 : \text { Find the Taylor polynomials } P_{1}(x), P_{2}(x), \text { and } P_{3}(x) \text { at }} \\\ {x=0 .}\\\\{\text { Step } 3 : \text { Calculate the }(n+1) \text { st derivative } f^{(n+1)}(c) \text { associated }} \\ {\text { with the remainder term for each Taylor polynomial. }} \\ {\text { Plot the derivative as a function of } c \text { over the specified interval }} \\ {\text { and estimate its maximum absolute value, } M .}\\\\{\text { Step } 4 : \text { Calculate the remainder } R_{n}(x) \text { for each polynomial. }} \\ {\text { Using the estimate } M \text { from Step } 3 \text { in place of } f^{(n+1)}(c), \text { plot }} \\ {R_{n}(x) \text { over the specified interval. Then estimate the values of }} \\ {x \text { that answer question (a). }}\\\\{\text { Step } 5 : \text { Compare your estimated error with the actual error }} \\ {E_{n}(x)=\left|f(x)-P_{n}(x)\right| \text { by plotting } E_{n}(x) \text { over the specified }} \\ {\text { interval. This will help answer question (b). }} \\ {\text { Step } 6 : \text { Graph the function and its three Taylor approximations }} \\ {\text { together. Discuss the graphs in relation to the information }} \\ {\text { discovered in Steps } 4 \text { and } 5 .}\end{array} \end{equation} $$f(x)=(1+x)^{3 / 2},-\frac{1}{2} \leq x \leq 2$$

Uniqueness of convergent power series a. Show that if two power series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) and \(\sum_{n=0}^{\infty} b_{n} x^{n}\) are convergent and equal for all values of \(x\) in an open interval \((-c, c),\) then \(a_{n}=b_{n}\) for every \(n\) . (Hint: Let \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}=\sum_{n=0}^{\infty} b_{n} x^{n} .\) Differentiate term by term to show that \(a_{n}\) and \(b_{n}\) both equal \(f^{(n)}(0) /(n !) . )\) b. Show that if \(\sum_{n=0}^{\infty} a_{n} x^{n}=0\) for all \(x\) in an open interval \((-c, c),\) then \(a_{n}=0\) for every \(n .\)

The Taylor series generated by \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is \(\sum_{n=0}^{\infty} a_{n} x^{n}\) A function defined by a power series \(\Sigma_{n=0}^{\infty} a_{n} x^{n}\) with a radius of convergence \(R>0\) has a Taylor series that converges to the function at every point of \((-R, R) .\) Show this by showing that the Taylor series generated by \(f(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) is the series \(\sum_{n=0}^{\infty} a_{n} x^{n}\) itself. \begin{equation} \begin{array}{c}{\text { An immediate consequence of this is that series like }} \\ {x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}+\cdots} \\ {\text {and}}\end{array} \end{equation} \begin{equation} x^{2} e^{x}=x^{2}+x^{3}+\frac{x^{4}}{2 !}+\frac{x^{5}}{3 !}+\cdots \end{equation} obtained by multiplying Taylor series by powers of \(x,\) as well as series obtained by integration and differentiation of convergent power series, are themselves the Taylor series generated by the functions they represent.

Use the Taylor series for 1\(/\left(1-x^{2}\right)\) to obtain a series for 2\(x /\left(1-x^{2}\right)^{2}\)
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