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Chapter 8: Problem 1

Match the formulas with the descriptions of the behavior of the sequence as \(n \rightarrow \infty\). 1\. \(s_{n}=n(n+1)-1\) 2\. \(s_{n}=1 /(n+1)\) 3\. \(s_{n}=3-1 / n\) 4\. \(s_{n}=n \sin (n) /(n+1)\) 5\. \(s_{n}=(n+1) / n\) A. does not converge, but doesn'\operatorname{tg} o ~ t o ~ \(\pm \infty\) B. converges to three from below C. diverges to \(\infty\) D. converges to one from above E. converges to zero through positive numbers

Short Answer

Expert verified
1. C, 2. E, 3. B, 4. A, 5. D.

Step by step solution

01

Match the first formula

Consider the formula for sequence 1: \[ s_{n} = n(n+1) - 1 \] As \( n \rightarrow \infty \), the term \( n(n+1) \) grows quadratically, hence \[ s_{n} \rightarrow \infty \] Therefore, the sequence diverges to infinity. The matching option is C.
02

Match the second formula

Consider the formula for sequence 2: \[ s_{n} = \frac{1}{n+1} \] As \( n \rightarrow \infty \), the denominator grows large, making the overall fraction approach zero: \[ s_{n} \rightarrow 0^{+} \] The sequence converges to zero through positive numbers. The matching option is E.
03

Match the third formula

Consider the formula for sequence 3: \[ s_{n} = 3 - \frac{1}{n} \] As \( n \rightarrow \infty \), \( \frac{1}{n} \) approaches zero, making \[ s_{n} \rightarrow 3 \] Since \( s_{n} \) approaches 3 from below, the matching option is B.
04

Match the fourth formula

Consider the formula for sequence 4: \[ s_{n}=\frac{n \sin(n)}{n+1} \] Given that \( n \sin(n) \) oscillates between \( -n \) and \( n \), the sequence does not settle into a single value but also does not diverge to \( \infty \). Therefore, it does not converge but also doesn鈥檛 go to \( \pm \infty \). The matching option is A.
05

Match the fifth formula

Consider the formula for sequence 5: \[ s_{n} = \frac{n+1}{n} \] As \( n \rightarrow \infty \), \[ s_{n} = 1 + \frac{1}{n} \] and \( \frac{1}{n} \) approaches zero, so \[ s_{n} \rightarrow 1^{+} \] The sequence converges to one from above. The matching option is D.

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Key Concepts

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

Convergence
When we talk about convergence in sequences, we refer to how the terms of the sequence behave as the index (usually denoted as 'n') approaches infinity. A sequence converges if its terms get closer and closer to a specific value, known as the limit, as n gets larger. This specific value is where the sequence 'settles'.
Some key points on convergence:
  • A sequence \( s_n \) converges to a limit L if for every small positive number \( \varepsilon \), there is a corresponding large number N such that for all n \> N, \( |s_n - L| < \varepsilon \).
  • Sequences can converge from above or below, as seen in examples like \( s_n = 3 - \frac{1}{n} \) converging to 3 from below and \( s_n = \frac{n+1}{n} \) converging to 1 from above.
Understanding convergence helps us make sense of sequences that appear to 'settle down' to some value as we go further out into the sequence.
Divergence
Divergence in sequences is, in many ways, the opposite of convergence. A sequence is said to diverge if its terms do not approach a specific limit as the index n approaches infinity. Instead of getting closer to a particular value, the terms either increase or decrease without bound, or they oscillate without settling on a particular value.
Key observations about divergence include:
  • If a sequence grows indefinitely, it diverges to \( \infty \) or \( -\infty \). For example, \( s_n = n(n+1) - 1 \) diverges to infinity.
  • Some sequences do not settle into any particular value nor do they grow without bound; they simply oscillate. An example is \( s_n = \frac{n \sin(n)}{n+1} \), which neither converges nor diverges to infinity.
Recognizing divergence helps in understanding sequences that do not fit the neat behavior of converging to a single value.
Limits
The limit of a sequence defines the value that the terms of the sequence approach as the index n tends to infinity. Limits are a fundamental concept in calculus and are vital in understanding the long-term behavior of sequences.
Important aspects of limits include:
  • Formally, a sequence \( s_n \) has a limit L if for every \( \varepsilon > 0 \), there exists an integer N such that for all \( n \geq N \, |s_n - L| < \varepsilon \).
  • In the sequence \( s_n = \frac{1}{n+1} \), as n approaches infinity, \( s_n \) approaches 0. Therefore, the limit of this sequence is 0.
  • Limits can be finite, like the limit of \( s_n = 3 - \frac{1}{n} \), which is 3, or infinite, where the sequence grows without bounds.
Understanding limits helps in determining the end-behavior of functions and sequences, simplifying complex mathematical problems by focusing on their behavior at infinity.

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Most popular questions from this chapter

Conditionally convergent series exhibit interesting and unexpected behavior. In this exercise we examine the conditionally convergent alternating harmonic series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\) and discover that addition is not commutative for conditionally convergent series. We will also encounter Riemann's Theorem concerning rearrangements of conditionally convergent series. Before we begin, we remind ourselves that $$ \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}=\ln (2) $$ a fact which will be verified in a later section.a. First we make a quick analysis of the positive and negative terms of the alternating harmonic series. i. Show that the series \(\sum_{k=1}^{\infty} \frac{1}{2 k}\) diverges. ii. Show that the series \(\sum_{k=1}^{\infty} \frac{1}{2 k+1}\) diverges. iii. Based on the results of the previous parts of this exercise, what can we say about the sums \(\sum_{k=C}^{\infty} \frac{1}{2 k}\) and \(\sum_{k=C}^{\infty} \frac{1}{2 k+1}\) for any positive integer \(C ?\) Be specific in your explanation. b. Recall addition of real numbers is commutative; that is $$ a+b=b+a $$ for any real numbers \(a\) and \(b\). This property is valid for any sum of finitely many terms, but does this property extend when we add infinitely many terms together? The answer is no, and something even more odd happens. Riemann's Theorem (after the nineteenth-century mathematician Georg Friedrich Bernhard Riemann) states that a conditionally convergent series can be rearranged to converge to any prescribed sum. More specifically, this means that if we choose any real number \(S\), we can rearrange the terms of the alternating harmonic series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\) so that the sum is \(S\). To understand how Riemann's Theorem works, let's assume for the moment that the number \(S\) we want our rearrangement to converge to is positive. Our job is to find a way to order the sum of terms of the alternating harmonic series to converge to \(S\). i. Explain how we know that, regardless of the value of \(S\), we can find a partial sum \(P_{1}\) $$ P_{1}=\sum_{k=1}^{n_{1}} \frac{1}{2 k+1}=1+\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2 n_{1}+1} $$ of the positive terms of the alternating harmonic series that equals or exceeds \(S\). Let $$ S_{1}=P_{1} $$ii. Explain how we know that, regardless of the value of \(S_{1}\), we can find a partial sum \(N_{1}\) $$ N_{1}=-\sum_{k=1}^{m_{1}} \frac{1}{2 k}=-\frac{1}{2}-\frac{1}{4}-\frac{1}{6}-\cdots-\frac{1}{2 m_{1}} $$ so that $$ S_{2}=S_{1}+N_{1} \leq S $$ iii. Explain how we know that, regardless of the value of \(S_{2}\), we can find a partial sum \(P_{2}\) $$ P_{2}=\sum_{k=n_{1}+1}^{n_{2}} \frac{1}{2 k+1}=\frac{1}{2\left(n_{1}+1\right)+1}+\frac{1}{2\left(n_{1}+2\right)+1}+\cdots+\frac{1}{2 n_{2}+1} $$ of the remaining positive terms of the alternating harmonic series so that $$ S_{3}=S_{2}+P_{2} \geq S $$iv. Explain how we know that, regardless of the value of \(S_{3}\), we can find a partial sum $$ N_{2}=-\sum_{k=m_{1}+1}^{m_{2}} \frac{1}{2 k}=-\frac{1}{2\left(m_{1}+1\right)}-\frac{1}{2\left(m_{1}+2\right)}-\cdots-\frac{1}{2 m_{2}} $$ of the remaining negative terms of the alternating harmonic series so that $$ S_{4}=S_{3}+N_{2} \leq S $$ v. Explain why we can continue this process indefinitely and find a sequence \(\left\\{S_{n}\right\\}\) whose terms are partial sums of a rearrangement of the terms in the alternating harmonic series so that \(\lim _{n \rightarrow \infty} S_{n}=S\).

For the following alternating series, \(\sum_{n=1}^{\infty} a_{n}=1-\frac{1}{10}+\frac{1}{100}-\frac{1}{1000}+\ldots\) how many terms do you have to go for your approximation (your partial sum) to be within 1e-07 from the convergent value of that series?

Finding limits of convergent sequences can be a challenge. However, there is a useful tool we can adapt from our study of limits of continuous functions at infinity to use to find limits of sequences. We illustrate in this exercise with the example of the sequence $$ \frac{\ln (n)}{n} $$ a. Calculate the first 10 terms of this sequence. Based on these calculations, do you think the sequence converges or diverges? Why? b. For this sequence, there is a corresponding continuous function \(f\) defined by $$ f(x)=\frac{\ln (x)}{x} $$ Draw the graph of \(f(x)\) on the interval [0,10] and then plot the entries of the sequence on the graph. What conclusion do you think we can draw about the sequence \(\left\\{\frac{\ln (n)}{n}\right\\}\) if \(\lim _{x \rightarrow \infty} f(x)=L ?\) Explain. c. Note that \(f(x)\) has the indeterminate form \(\frac{\infty}{\infty}\) as \(x\) goes to infinity. What idea from differential calculus can we use to calculate \(\lim _{x \rightarrow \infty} f(x) ?\) Use this method to find \(\lim _{x \rightarrow \infty} f(x) .\) What, then, is \(\lim _{n \rightarrow \infty} \frac{\ln (n)}{n} ?\)

The associative and distributive laws of addition allow us to add finite sums in any order we want. That is, if \(\sum_{k=0}^{n} a_{k}\) and \(\sum_{k=0}^{n} b_{k}\) are finite sums of real numbers, then $$ \sum_{k=0}^{n} a_{k}+\sum_{k=0}^{n} b_{k}=\sum_{k=0}^{n}\left(a_{k}+b_{k}\right) $$ However, we do need to be careful extending rules like this to infinite series. a. Let \(a_{n}=1+\frac{1}{2^{n}}\) and \(b_{n}=-1\) for each nonnegative integer \(n\). \- Explain why the series \(\sum_{k=0}^{\infty} a_{k}\) and \(\sum_{k=0}^{\infty} b_{k}\) both diverge. \- Explain why the series \(\sum_{k=0}^{\infty}\left(a_{k}+b_{k}\right)\) converges. \- Explain why $$ \sum_{k=0}^{\infty} a_{k}+\sum_{k=0}^{\infty} b_{k} \neq \sum_{k=0}^{\infty}\left(a_{k}+b_{k}\right) $$ This shows that it is possible to have to two divergent series \(\sum_{k=0}^{\infty} a_{k}\) and \(\sum_{k=0}^{\infty} b_{k}\) but yet have the series \(\sum_{k=0}^{\infty}\left(a_{k}+b_{k}\right)\) converge. b. While part (a) shows that we cannot add series term by term in general, we can under reasonable conditions. The problem in part (a) is that we tried to add divergent series. In this exercise we will show that if \(\sum a_{k}\) and \(\sum b_{k}\) are convergent series, then \(\sum\left(a_{k}+b_{k}\right)\) is a convergent series and $$ \sum\left(a_{k}+b_{k}\right)=\sum a_{k}+\sum b_{k} $$ - Let \(A_{n}\) and \(B_{n}\) be the \(n\) th partial sums of the series \(\sum_{k=1}^{\infty} a_{k}\) and \(\sum_{k=1}^{\infty} b_{k}\), respectively. Explain why $$ A_{n}+B_{n}=\sum_{k=1}^{n}\left(a_{k}+b_{k}\right) $$ \- Use the previous result and properties of limits to show that $$ \sum_{k=1}^{\infty}\left(a_{k}+b_{k}\right)=\sum_{k=1}^{\infty} a_{k}+\sum_{k=1}^{\infty} b_{k} . $$ (Note that the starting point of the sum is irrelevant in this problem, so it doesn't matter where we begin the sum.) c. Use the prior result to find the sum of the series \(\sum_{k=0}^{\infty} \frac{2^{k}+3^{k}}{5^{k}}\).

In the Limit Comparison Test we compared the behavior of a series to one whose behavior we know. In that test we use the limit of the ratio of corresponding terms of the series to determine if the comparison is valid. In this exercise we see how we can compare two series directly, term by term, without using a limit of sequence. First we consider an example. a. Consider the series \(\sum \frac{1}{k^{2}}\) and \(\sum \frac{1}{k^{2}+k}\) We know that the series \(\sum \frac{1}{k^{2}}\) is a \(p\) -series with \(p=2>1\) and so \(\sum \frac{1}{k^{2}}\) converges. In this part of the exercise we will see how to use information about \(\sum \frac{1}{k^{2}}\) to determine information about \(\sum \frac{1}{k^{2}+k} .\) Let \(a_{k}=\frac{1}{k^{2}}\) and \(b_{k}=\frac{1}{k^{2}+k} .\) i) Let \(S_{n}\) be the \(n\) th partial sum of \(\sum \frac{1}{k^{2}}\) and \(T_{n}\) the \(n\) th partial sum of \(\sum \frac{1}{k^{2}+k}\). Which is larger, \(S_{1}\) or \(T_{1}\) ? Why? ii) Recall that $$ S_{2}=S_{1}+a_{2} \text { and } T_{2}=T_{1}+b_{2} $$ Which is larger, \(a_{2}\) or \(b_{2}\) ? Based on that answer, which is larger, \(S_{2}\) or \(T_{2}\) ? iii) Recall that $$ S_{3}=S_{2}+a_{3} \text { and } T_{3}=T_{2}+b_{3} $$ Which is larger, \(a_{3}\) or \(b_{3}\) ? Based on that answer, which is larger, \(S_{3}\) or \(T_{3}\) ? iv) Which is larger, \(a_{n}\) or \(b_{n}\) ? Explain. Based on that answer, which is larger, \(S_{n}\) or \(T_{n}\) ? v) Based on your response to the previous part of this exercise, what relationship do you expect there to be between \(\sum \frac{1}{k^{2}}\) and \(\sum \frac{1}{k^{2}+k} ?\) Do you expect \(\sum \frac{1}{k^{2}+k}\) to converge or diverge? Why? b. The example in the previous part of this exercise illustrates a more general result. Explain why the Direct Comparison Test, stated here, works.
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