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Chapter 8: Problem 34

Write the terms \(a_{1}, a_{2}, a_{3},\) and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. $$a_{n+1}=\frac{a_{n}}{10} ; a_{0}=1$$

Short Answer

Expert verified
Answer: The first four terms of the sequence are \(a_1 = 0.1, a_2 = 0.01, a_3 = 0.001\), and \(a_4 = 0.0001\). The sequence converges as the terms get smaller and approach 0.

Step by step solution

01

Calculate \(a_1\) using \(a_0 = 1\)

Using the recursive formula, we find \(a_1\) from \(a_0\): $$a_1 = \frac{a_0}{10} = \frac{1}{10} = 0.1$$
02

Calculate \(a_2\) using \(a_1 = 0.1\)

Applying the recursive formula again, we find \(a_2\) from \(a_1\): $$a_2 = \frac{a_1}{10} = \frac{0.1}{10} = 0.01$$
03

Calculate \(a_3\) using \(a_2 = 0.01\)

Once more, apply the recursive formula to find \(a_3\) from \(a_2\): $$a_3 = \frac{a_2}{10} = \frac{0.01}{10} = 0.001$$
04

Calculate \(a_4\) using \(a_3 = 0.001\)

Lastly, find \(a_4\) from \(a_3\) using the recursive formula: $$a_4 = \frac{a_3}{10} = \frac{0.001}{10} = 0.0001$$
05

Determine convergence or divergence

Looking at the calculated terms (\(0.1, 0.01, 0.001, 0.0001\)), we can observe that the terms are getting smaller and smaller, approaching 0. Since the numbers get closer and closer to 0, the sequence converges. We can make a conjecture that the limit of the sequence as \(n\) approaches infinity is 0. This means if we continue finding the terms of the sequence, they will become exceedingly close to 0, but never equal to 0.

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Key Concepts

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

Recursive Sequences
Recursive sequences are mathematical sequences in which each term after the first is generated by applying a fixed rule to the preceding term(s). The defining characteristic of a recursive sequence is that you need to know one or more initial terms and the rule used to advance from one term to the next. For example, in the exercise provided, the rule is to divide the previous term by 10, starting with an initial term of 1. To determine subsequent terms, simply apply the rule sequentially.

As with any recursive sequence, understanding the pattern is crucial for determining the general behavior of the sequence, such as whether it converges or diverges.
Limits of Sequences
The limit of a sequence is a fundamental concept that describes the value that the terms of the sequence approach as the number of terms increases indefinitely. The limit does not always exist, but if it does, the sequence is said to converge to that limit. In our exercise, by continuously dividing by 10, the terms approach zero. Mathematically, this is expressed as \(\lim_{n\to\infty} a_n = 0\text{, where}\) \(a_n\text{ are the terms of the sequence.}\) If the terms of a sequence do not approach any particular value, the sequence diverges and does not have a limit.
Convergence of Sequences
A sequence converges when its terms tend toward a specific number, known as the limit of the sequence, as the number of terms grows. The convergence is often determined by the nature of the rule defining the sequence. For instance, in the given exercise, since each term is generated by dividing the previous term by 10, a positive number smaller than 1, it intuitively makes sense that the numbers get increasingly closer to 0.

The mathematical notation \(\lim_{n\to\infty} a_n = L\) is used, where \(L\) is the limit, to indicate that as \(n\), the term number, gets very large, the sequence's terms get arbitrarily close to \(L\), hence exhibiting convergence.
Divergence of Sequences
In contrast, a sequence diverges if its terms do not approach a specific value as the sequence progresses. Divergence can happen in several ways: the terms might continue to grow without bound, oscillate between values without settling, or even have multiple sub-sequences converging to different limits. If no single value can be identified that the terms of the sequence approach, the sequence is considered divergent. In mathematical terms, divergence means that there is no real number \(L\) such that \(\lim_{n\to\infty} a_n = L\). Divergence is an important concept to understand, as it indicates that the behavior of a sequence does not stabilize as it progresses.

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Most popular questions from this chapter

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=\sqrt{n} \text { and } b_{n}=2 \ln n, n \geq 3$$

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about \(\mathrm{A.D.} 1200\) as a model for the growth of rabbit populations. It is given by the recurrence relation \(f_{n+1}=f_{n}+f_{n-1},\) for \(n=1,2,3, \ldots,\) where \(f_{0}=1, f_{1}=1 .\) Each term of the sequence is the sum of its two predecessors. a. Write out the first ten terms of the sequence. b. Is the sequence bounded? c. Estimate or determine \(\varphi=\lim _{n \rightarrow \infty} \frac{f_{n+1}}{f_{n}},\) the ratio of the successive terms of the sequence. Provide evidence that \(\varphi=(1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Use induction to verify the remarkable result that $$f_{n}=\frac{1}{\sqrt{5}}\left(\varphi^{n}-(-1)^{n} \varphi^{-n}\right).$$

Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{n^{1000}}{2^{n}}\right\\}$$

Pick two positive numbers \(a_{0}\) and \(b_{0}\) with \(a_{0}>b_{0},\) and write out the first few terms of the two sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}:\) $$a_{n+1}=\frac{a_{n}+b_{n}}{2}, \quad b_{n+1}=\sqrt{a_{n} b_{n}}, \quad \text { for } n=0,1,2 \dots$$ (Recall that the arithmetic mean \(A=(p+q) / 2\) and the geometric mean \(G=\sqrt{p q}\) of two positive numbers \(p\) and \(q\) satisfy \(A \geq G.)\) a. Show that \(a_{n} > b_{n}\) for all \(n\). b. Show that \(\left\\{a_{n}\right\\}\) is a decreasing sequence and \(\left\\{b_{n}\right\\}\) is an increasing sequence. c. Conclude that \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) converge. d. Show that \(a_{n+1}-b_{n+1}<\left(a_{n}-b_{n}\right) / 2\) and conclude that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) The common value of these limits is called the arithmetic-geometric mean of \(a_{0}\) and \(b_{0},\) denoted \(\mathrm{AGM}\left(a_{0}, b_{0}\right)\). e. Estimate AGM(12,20). Estimate Gauss' constant \(1 / \mathrm{AGM}(1, \sqrt{2})\).

Imagine that the government of a small community decides to give a total of \(\$ W\), distributed equally, to all its citizens. Suppose that each month each citizen saves a fraction \(p\) of his or her new wealth and spends the remaining \(1-p\) in the community. Assume no money leaves or enters the community, and all the spent money is redistributed throughout the community. a. If this cycle of saving and spending continues for many months, how much money is ultimately spent? Specifically, by what factor is the initial investment of \(\$ W\) increased (in terms of \(p\) )? Economists refer to this increase in the investment as the multiplier effect. b. Evaluate the limits \(p \rightarrow 0\) and \(p \rightarrow 1,\) and interpret their meanings.
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