Textbook Question
Find each indicated sum.
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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 37
Find the sum of each infinite geometric series. 1 + 1/3 + 1/9 + 1/27 + ...
Verified step by step guidance1
Identify the first term \( a \) of the infinite geometric series. In this series, the first term is \( 1 \).
Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{1/3}{1} = \frac{1}{3} \).
Verify that the absolute value of the common ratio is less than 1, i.e., \( |r| < 1 \), which ensures the series converges. Here, \( |\frac{1}{3}| < 1 \), so the series converges.
Use the formula for the sum \( S \) of an infinite geometric series: \[ S = \frac{a}{1 - r} \], where \( a \) is the first term and \( r \) is the common ratio.
Substitute the values of \( a = 1 \) and \( r = \frac{1}{3} \) into the formula to express 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 the sum of infinitely many terms where each term is found by multiplying the previous term by a constant ratio. It has the form a + ar + ar² + ar³ + ..., where |r| < 1 for the series to converge.
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Common Ratio
The common ratio (r) is the factor by which each term in a geometric series is multiplied to get the next term. It is found by dividing any term by its preceding term. For convergence in infinite series, the absolute value of r must be less than 1.
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Sum Formula for Infinite Geometric Series
The sum S of an infinite geometric series with first term a and common ratio r (|r| < 1) is given by S = a / (1 - r). This formula allows calculation of the total sum without adding infinitely many terms.
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