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Chapter 9: Problem 1

What is the defining characteristic of a geometric series? Give an example.

Short Answer

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Short Answer: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant value called the common ratio. The general form is a, ar, ar^2, ar^3, ..., where 'a' is the first term and 'r' is the common ratio. For example, a geometric series with a first term of 2 and a common ratio of 3 would be: 2, 6, 18, 54, 162, ...

Step by step solution

01

Definition of Geometric Series

A geometric series is a sequence of numbers in which the quotient, also known as the common ratio, of any two consecutive terms is constant. In other words, every term after the first one is found by multiplying the previous term by the same constant value.
02

General Form

The general form of a geometric series is given by: a, ar, ar^2, ar^3, ... , where 'a' is the first term, 'r' is the common ratio, and each term is found by multiplying the previous term by 'r'.
03

Example of Geometric Series

Let's create a geometric series with a first term 'a' of 2 and a common ratio 'r' of 3. We will show the first 5 terms: 1st term, a: 2 2nd term, ar: 2 * 3 = 6 3rd term, ar^2: 6 * 3 = 18 4th term, ar^3: 18 * 3 = 54 5th term, ar^4: 54 * 3 = 162 Hence, the example of a geometric series is: 2, 6, 18, 54, 162, ...

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Most popular questions from this chapter

Infinite products Use the ideas of Exercise 88 to evaluate the following infinite products. $$\text { a. } \prod_{k=0}^{\infty} e^{1 / 2^{k}}=e \cdot e^{1 / 2} \cdot e^{1 / 4} \cdot e^{1 / 8} \ldots$$ $$\text { b. } \prod_{k=2}^{\infty}\left(1-\frac{1}{k}\right)=\frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \dots$$

a. Consider the number 0.555555...., which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 \ldots\) b. Consider the number \(0.54545454 \ldots,\) which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots \ldots, n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form of the number. d. Try the method of part (c) on the number \(0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{\cos k}{k^{3}}$$

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about 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 \(f_{0}=0, f_{1}=1 .\) 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. Verify the remarkable result that $$f_{n}=\frac{1}{\sqrt{5}}\left(\varphi^{n}-(-1)^{n} \varphi^{-n}\right)$$

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty}(-1)^{k}$$
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