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Chapter 9: Problem 40

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. $$a_{n+1}=10 a_{n}-1 ; a_{0}=0$$

Short Answer

Expert verified
Question: Determine whether the sequence given by the recursive formula \(a_{n+1} = 10a_n - 1 \) with the initial term \(a_0 = 0\) converges or diverges. Provide the first four terms of the sequence. Answer: The sequence diverges, and the first four terms are -1, -11, -111, and -1111.

Step by step solution

01

Find the first term a_1

Using the given recursive formula and initial value, we have: \(a_1 = 10a_0 - 1 = 10(0) - 1 = -1\)
02

Find the second term a_2

Using the recursive formula and the value of \(a_1\), we have: \(a_2 = 10a_1 - 1 = 10(-1) - 1 = -11\)
03

Find the third term a_3

Using the recursive formula and the value of \(a_2\), we have: \(a_3 = 10a_2 - 1 = 10(-11) - 1 = -111\)
04

Find the fourth term a_4

Using the recursive formula and the value of \(a_3\), we have: \(a_4 = 10a_3 - 1 = 10(-111) - 1 = -1111\)
05

Determine if the sequence converges or diverges

Looking at the first four terms of the sequence, we can notice a pattern: $$-1, -11, -111, -1111, \dots$$ The magnitudes of the terms are increasing, meaning the sequence isn't approaching a particular value. Consequently, the sequence diverges. The reason behind the divergence is that each new term is generated by multiplying the previous term by 10 and then subtracting 1. The multiplication by 10 makes the magnitude of the terms grow without bound, so the sequence doesn't converge to any limit.

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Most popular questions from this chapter

Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than \(10^{-4} .\) Although you do not need it, the exact value of the series is given in each case. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(2 k+1) !}$$

Evaluate the limit of the following sequences. $$a_{n}=\tan ^{-1}\left(\frac{10 n}{10 n+4}\right)$$

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{k}}{k !}$$

Consider the sequence \(\left\\{x_{n}\right\\}\) defined for \(n=1,2,3, \ldots\) by $$x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2 n}$$ a. Write out the terms \(x_{1}, x_{2}, x_{3}\) b. Show that \(\frac{1}{2} \leq x_{n}<1,\) for \(n=1,2,3, \ldots\) c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using \(n\) subintervals. d. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\)

Infinite products An infinite product \(P=a_{1} a_{2} a_{3} \ldots,\) which is denoted \(\prod_{k=1}^{\infty} a_{k}\) is the limit of the sequence of partial products \(\left\\{a_{1}, a_{1} a_{2}, a_{1} a_{2} a_{3}, \dots\right\\}\) a. Show that the infinite product converges (which means its sequence of partial products converges) provided the series \(\sum_{k=1}^{\infty} \ln a_{k}\) converges. b. Consider the infinite product $$P=\prod_{k=2}^{\infty}\left(1-\frac{1}{k^{2}}\right)=\frac{3}{4} \cdot \frac{8}{9} \cdot \frac{15}{16} \cdot \frac{24}{25} \cdots$$ Write out the first few terms of the sequence of partial products, $$P_{n}=\prod_{k=2}^{n}\left(1-\frac{1}{k^{2}}\right)$$ (for example, \(P_{2}=\frac{3}{4}, P_{3}=\frac{2}{3}\) ). Write out enough terms to determine the value of the product, which is \(\lim _{n \rightarrow \infty} P_{n}\). c. Use the results of parts (a) and (b) to evaluate the series $$\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right)$$
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