/*! 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 38 Find the sum of each infinite ge... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 38

Find the sum of each infinite geometric series, if it exists. \(1-0.5+0.25-\ldots\)

Short Answer

Expert verified
The sum of the series is \( \frac{2}{3} \).

Step by step solution

01

Identify the First Term

In a geometric series, the first term is denoted by \( a \). In the series \(1 - 0.5 + 0.25 - \ldots\), the first term \( a \) is 1.
02

Find the Common Ratio

The common ratio \( r \) is found by dividing the second term by the first term. For this series, \( r = \frac{-0.5}{1} = -0.5 \).
03

Check the Convergence Condition

The sum of an infinite geometric series exists only if the common ratio \( r \) satisfies \(|r| < 1\). Here, \(|-0.5| = 0.5 < 1\), so the series converges.
04

Use the Infinite Series Sum Formula

The sum \( S \) of an infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term, and \( r \) is the common ratio.
05

Calculate the Sum

Substitute the values for \( a \) and \( r \) into the formula: \( S = \frac{1}{1 - (-0.5)} = \frac{1}{1 + 0.5} = \frac{1}{1.5} = \frac{2}{3} \).

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!

Key Concepts

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

Infinite Series
An infinite series is essentially the sum of an unlimited number of terms in a sequence. Think of it as a way to add numbers forever! However, not every infinite series leads to a neat, finite sum. In mathematics, some infinite series grow endlessly, while others settle into a specific value. These outcomes depend heavily on the structure or rules of the series.
  • An infinite geometric series, the focus of our exercise, repeatedly multiplies a starting value by a fixed number, known as the common ratio.
  • Our series, for instance, starts at 1 and keeps getting multiplied by \(-0.5\): \(1, -0.5, 0.25, \ldots\).
Recognizing a pattern here is key to understanding whether or how the series will sum up to something simple.
Convergence Condition
For an infinite geometric series to have a finite sum, it must satisfy a specific condition known as the convergence condition. This condition hinges on the common ratio, which must have a magnitude less than one.
  • This means the series keeps multiplying by a smaller and smaller factor as it goes on.
  • Mathematically, the series converges if \(|r| < 1\), where \(r\) represents the common ratio.
In our example, the common ratio is -0.5. Since \(|-0.5| = 0.5 < 1\), the series meets the convergence condition, confirming that a specific sum can be found. Otherwise, the terms would grow or oscillate indefinitely, making them impossible to neatly sum together.
Common Ratio
The common ratio is the secret ingredient in a geometric series. It tells you how to get from one term to the next. You find it by dividing any term by the one before it.
  • In our series \(1, -0.5, 0.25, \ldots\), dividing -0.5 by 1 gives -0.5, and dividing 0.25 by -0.5 also gives -0.5.
  • This consistency confirms that our chosen series is indeed geometric.
Such a common ratio shapes the behavior and value of the series. If it's too large (\

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 each expression. $$ \frac{9 !}{7 !} $$

PUZZLES Show that a \(2^{n}\) by \(2^{n}\) checkerboard with the top right square missing can always be covered by nonoverlapping L-shaped tiles like the one at the right.

Find the first five terms of each sequence. $$ a_{1}=3, a_{n+1}=2 a_{n}-1 $$

A wooden pole swings back and forth over the cup on a miniature golf hole. One player pulls the pole to the side and lets it go. Then it follows a swing pattern of 25 centimeters, 20 centimeters, 16 centimeters, and so on until it comes to rest. What is the total distance the pole swings before coming to rest?

State whether each statement is true or false when \(n=1\). Explain. $$ 1=\frac{(n+1)(2 n+1)}{2} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

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.