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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.R.37
Chapter 10, Problem 10.R.37

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 1 to ∞)2ᵏ / 3ᵏ⁺²

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1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{2^k}{3^{k+2}} \). Notice that the series involves terms with exponents depending on \(k\).
Rewrite the general term to separate the powers of 3: \( \frac{2^k}{3^{k+2}} = \frac{2^k}{3^k \cdot 3^2} = \frac{1}{3^2} \cdot \frac{2^k}{3^k} = \frac{1}{9} \cdot \left( \frac{2}{3} \right)^k \).
Recognize that the series is a geometric series with the first term \( a = \frac{1}{9} \cdot \left( \frac{2}{3} \right)^1 = \frac{2}{27} \) and common ratio \( r = \frac{2}{3} \).
Check the convergence of the geometric series by verifying if \( |r| < 1 \). Since \( \frac{2}{3} < 1 \), the series converges.
Use the formula for the sum of an infinite geometric series starting at \( k=1 \): \[ S = a \cdot \frac{1}{1 - r} \], where \( a \) is the first term and \( r \) is the common ratio. Substitute the values to express the sum.

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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 a sum of terms where each term is found by multiplying the previous term by a constant ratio. It converges if the absolute value of the ratio is less than 1, and its sum can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio.
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Convergence and Divergence of Series

A series converges if the sum of its infinite terms approaches a finite limit; otherwise, it diverges. For geometric series, convergence depends on the common ratio's magnitude. Understanding convergence is essential to determine whether the series sum exists or not.
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Convergence of an Infinite Series

Manipulating Series Terms

Simplifying the general term of a series often involves algebraic manipulation, such as factoring exponents or rewriting terms to identify the first term and common ratio. This step is crucial to apply known formulas and test for convergence effectively.
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Geometric Series
Related Practice
Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

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Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

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A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.


a.How far does the crew dig in 10 weeks? 20 weeks? N weeks?

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Textbook Question

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

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Determine whether the following series converge absolutely, converge conditionally, or diverge.

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