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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.3.19b
Chapter 10, Problem 10.3.19b

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.


b. Evaluate the series using Theorem 10.7.


∑ (k = 0 to ∞) (–2/7)ᵏ

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Identify the geometric series given: \(\sum_{k=0}^{\infty} \left(-\frac{2}{7}\right)^k\). Here, the first term \(a\) is when \(k=0\), which is \(\left(-\frac{2}{7}\right)^0 = 1\).
Recognize the common ratio \(r\) of the geometric series, which is the base of the exponent: \(r = -\frac{2}{7}\).
Check the convergence condition for an infinite geometric series: the series converges if and only if \(|r| < 1\). Since \(| -\frac{2}{7} | = \frac{2}{7} < 1\), the series converges.
Apply Theorem 10.7 for the sum of an infinite geometric series, which states that if \(|r| < 1\), then the sum \(S\) is given by \(S = \frac{a}{1 - r}\).
Substitute the values of \(a = 1\) and \(r = -\frac{2}{7}\) into the formula to express the sum as \(S = \frac{1}{1 - \left(-\frac{2}{7}\right)}\). Simplify the denominator to get the final expression for the sum.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. It can be finite or infinite, and its general form is ∑ ar^k, where a is the first term and r is the common ratio.
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Convergence of Infinite Geometric Series

An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. When it converges, the sum can be calculated using the formula S = a / (1 - r), providing a finite value for the infinite sum.
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Convergence of an Infinite Series

Theorem 10.7 (Sum of an Infinite Geometric Series)

Theorem 10.7 states that for an infinite geometric series with first term a and common ratio r, if |r| < 1, then the sum is S = a / (1 - r). This theorem provides a direct method to evaluate infinite geometric series without summing individual terms.
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Related Practice
Textbook Question

67–70. Formulas for sequences of partial sums Consider the following infinite series.


b.Find a formula for the nth partial sum Sₙ of the infinite series. Use this formula to find the next four partial sums S₅, S₆, S₇, S₈ of the infinite series.


∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Textbook Question

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


b. Find how many terms are needed to ensure that the remainder is less than 10⁻³.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

Textbook Question

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


b. Find an explicit formula for the nth term of the sequence {hₙ}.


h₀ = 20,r = 0.5

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).


{1, 2, 4, 8, 16, ......}

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. A series that converges absolutely must converge.

Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

{-5, 5, -5, 5, ......}

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