Textbook Question
Express each repeating decimal as a fraction in lowest terms.
Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 48
Find the sum of each infinite geometric series. -6 + 4 - 8/3 + 16/9 - ...
Verified step by step guidance1
Identify the first term \( a \) of the series. In this case, the first term is \( a = -6 \).
Determine the common ratio \( r \) by dividing the second term by the first term. For this series, \( r = \frac{4}{-6} = -\frac{2}{3} \). Verify this ratio holds for subsequent terms by dividing each term by the previous one.
Check if the series is convergent. An infinite geometric series converges if \( |r| < 1 \). Here, \( |r| = \frac{2}{3} < 1 \), so the series converges.
Use the formula for the sum of an infinite geometric series: \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio.
Substitute \( a = -6 \) and \( r = -\frac{2}{3} \) into the formula: \( S = \frac{-6}{1 - (-\frac{2}{3})} \). Simplify the denominator and calculate the result to find the sum of the series.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Geometric Series
An infinite geometric series is a sum of the terms of a geometric sequence that continues indefinitely. It is defined by a first term (a) and a common ratio (r). The series converges to a finite value if the absolute value of the common ratio is less than one (|r| < 1). The formula for the sum of an infinite geometric series is S = a / (1 - r).
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Common Ratio
The common ratio in a geometric series is the factor by which each term is multiplied to obtain the next term. It is calculated by dividing any term by its preceding term. For the series given, identifying the common ratio is crucial for applying the sum formula correctly. If the common ratio is greater than or equal to one in absolute value, the series diverges and does not have a finite sum.
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Convergence of Series
Convergence refers to the behavior of a series as the number of terms approaches infinity. An infinite series converges if the sum approaches a specific finite value. For geometric series, convergence is determined by the common ratio; if |r| < 1, the series converges. Understanding convergence is essential for determining whether the infinite series can be summed to a finite value.
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Related Practice
Textbook Question
Find the term indicated in each expansion. (x2 + y)22; the term containing y14
Textbook Question
Use the Binomial Theorem to expand each expression and write the result in simplified form. (x3 +x-2)4
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Textbook Question
Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+2+3+⋯+ 30
Textbook Question
Express each repeating decimal as a fraction in lowest terms.
Textbook Question
Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.