Domanda del libro di testo
Find each indicated sum.
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 9, Problema 37
Find the sum of each infinite geometric series. 1 + 1/3 + 1/9 + 1/27 + ...
Guida verificata passo dopo passo1
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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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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Geometric Sequences - Recursive Formula
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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5:57Graphs of Common Functions
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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4:18Geometric Sequences - Recursive Formula
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