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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 10.3.19a
Chapter 10, Problem 10.3.19a

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


a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.


∑ (k = 0 to ∞) (–2/7)ᵏ

Verified step by step guidance
1
Identify the first term \( a \) and the common ratio \( r \) of the geometric series. Here, \( a = 1 \) (since when \( k=0 \), \( (-2/7)^0 = 1 \)) and \( r = -\frac{2}{7} \).
Write the formula for the nth partial sum \( S_n \) of a geometric series: \[ S_n = a \frac{1 - r^{n+1}}{1 - r} \]
Substitute the values of \( a \) and \( r \) into the formula to express \( S_n \) explicitly: \[ S_n = 1 \times \frac{1 - \left(-\frac{2}{7}\right)^{n+1}}{1 - \left(-\frac{2}{7}\right)} \]
Evaluate the limit of \( S_n \) as \( n \to \infty \). Since \( |r| = \frac{2}{7} < 1 \), the term \( r^{n+1} \) approaches zero, so the limit is: \[ \lim_{n \to \infty} S_n = \frac{1}{1 - \left(-\frac{2}{7}\right)} \]
Simplify the expression for the limit to find the sum of the infinite geometric series.

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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 a sum of terms where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, where a is the first term and r is the common ratio. Understanding this structure is essential for finding partial sums and limits.
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Geometric Series

Partial Sum of a Geometric Series

The nth partial sum Sₙ of a geometric series is the sum of the first n+1 terms. It can be calculated using the formula Sₙ = a(1 - r^(n+1)) / (1 - r) when r ≠ 1. This formula helps in evaluating the series up to a finite number of terms.
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Intro to Series: Partial Sums

Limit of an Infinite Geometric Series

If the absolute value of the common ratio |r| < 1, the infinite geometric series converges, and its sum is the limit of the partial sums as n approaches infinity. This limit is given by S = a / (1 - r), providing the sum of infinitely many terms.
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Geometric Series
Related Practice
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.


a. Find the first four terms of the sequence of heights {hₙ}.


h₀ = 30,r = 0.25

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Textbook Question

39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.


a. Use Sₙ to estimate the sum of the series.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.

Textbook Question

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

a. Find the next two terms of the sequence.


{1, 3, 9, 27, 81, ......}

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Textbook Question

Find the first term a and the ratio r of each geometric series.


a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ

Textbook Question

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

a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.