/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Does a geometric series always h... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 5

Does a geometric series always have a finite value?

Short Answer

Expert verified
Explain. Answer: No, a geometric series does not always have a finite value. It converges and has a finite value if and only if the common ratio 'r' lies between -1 and 1, that is, -1 < r < 1. If the common ratio is greater than or equal to 1 or less than or equal to -1, the series will diverge and will not have a finite sum.

Step by step solution

01

Definition of a Geometric Series

A geometric series is the sum of the terms of a geometric sequence. A geometric sequence is a set of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). A geometric series can be represented as: S_n = a_1 + a_1 * r + a_1 * r^2 + a_1 * r^3 + ... + a_1 * r^(n-1), where n is the number of terms, and a_1 is the first term.
02

Convergence of a Geometric Series

A geometric series converges (i.e., has a finite value) if the sum of its terms approaches a finite limit as the number of terms increases. The convergence of a geometric series depends on the value of the common ratio 'r.' For a geometric series to converge, the common ratio 'r' must lie between -1 and 1, i.e., -1 < r < 1.
03

Formula for the Sum of a Convergent Geometric Series

If a geometric series converges, we can find the sum using the formula: S = a_1 * (1 - r^n) / (1 - r). In this formula, S represents the sum of the series, n is the number of terms, a_1 is the first term, and r is the common ratio. Notice that this formula is valid only when -1 < r < 1.
04

Divergence of a Geometric Series

If the common ratio 'r' is greater than or equal to 1, or less than or equal to -1, the geometric series will diverge, which means it will not have a finite sum. In this case, the series will either increase without bound or oscillate between two values.
05

Conclusion

A geometric series does not always have a finite value. It converges and has a finite value if and only if the common ratio 'r' lies between -1 and 1, i.e., -1 < r < 1. If the common ratio is greater than or equal to 1 or less than or equal to -1, the series will diverge and will not have a finite sum.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate the geometric series or state that it diverges. $$\sum_{k=1}^{\infty}(-e)^{-k}$$

Explain why the remainder in terminating an alternating series is less than or equal to the first neglected term.

Suppose a ball is thrown upward to a height of \(h_{0}\) meters. Each time the ball bounces, it rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce. Consider the following values of \(h_{0}\) and \(r.\) a. Find the first four terms of the sequence of heights \(\left\\{h_{n}\right\\}.\) b. Find an explicit formula for the nth term of the sequence \(\left\\{h_{n}\right\\}.\) $$h_{0}=10, r=0.9$$

Consider the following convergent series. a. Find an upper bound for the remainder in terms of \(n\) b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\) c. Find lower and upper bounds $$\left(L_{n} \text { and } U_{n}\right.$$ respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. $$\sum_{k=1}^{\infty} \frac{1}{k^{8}}$$

Suppose a ball is thrown upward to a height of \(h_{0}\) meters. Each time the ball bounces, it rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce and let \(S_{n}\) be the total distance the ball has traveled at the moment of the nth bounce. a. Find the first four terms of the sequence \(\left\\{S_{n}\right\\}\) b. Make a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine a plausible value for the limit of \(\left\\{S_{n}\right\\}.\) $$h_{0}=20, r=0.5$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.