Domanda del libro di testo
Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.
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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 41
Find the sum of each infinite geometric series. 1 - 1/2 + 1/4 - 1/8 + ...
Guida verificata passo dopo passo1
Identify the first term \( a \) of the infinite geometric series. In this series, the first term is \( a = 1 \).
Determine the common ratio \( r \) by dividing the second term by the first term: \( r = \frac{-\frac{1}{2}}{1} = -\frac{1}{2} \).
Check if the series converges by verifying that the absolute value of the common ratio is less than 1: \( |r| = \left| -\frac{1}{2} \right| = \frac{1}{2} < 1 \), so the series converges.
Use the formula for the sum \( S \) of an infinite geometric series: \[ S = \frac{a}{1 - r} \].
Substitute the values of \( a \) and \( r \) into the formula to express the sum: \[ S = \frac{1}{1 - \left(-\frac{1}{2}\right)} \].
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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. The series continues indefinitely, and its behavior depends on the common ratio's value.
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Geometric Sequences - Recursive Formula
Common Ratio
The common ratio 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 of an infinite series, the absolute value of the common ratio must be less than 1.
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Sum Formula for Infinite Geometric Series
If the absolute value of the common ratio r is less than 1, the sum S of the infinite geometric series with first term a is given by S = a / (1 - r). This formula allows calculation of the series' sum without adding infinitely many terms.
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4:18Geometric Sequences - Recursive Formula
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