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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.57
Chapter 10, Problem 10.R.57

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)5ᵏ / 2²ᵏ⁺¹

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{5^k}{2^{2k+1}} \). This is an infinite series where each term is \( a_k = \frac{5^k}{2^{2k+1}} \).
Rewrite the general term to simplify the expression. Notice that \( 2^{2k+1} = 2^{2k} \cdot 2^1 = 2 \cdot (2^2)^k = 2 \cdot 4^k \). So, \( a_k = \frac{5^k}{2 \cdot 4^k} = \frac{1}{2} \cdot \left( \frac{5}{4} \right)^k \).
Recognize that the series is a geometric series with common ratio \( r = \frac{5}{4} \) and first term \( a_1 = \frac{1}{2} \cdot \left( \frac{5}{4} \right)^1 \).
Recall the convergence criterion for a geometric series: it converges if and only if \( |r| < 1 \). Here, check the absolute value of the ratio \( |r| = \frac{5}{4} \).
Since \( |r| > 1 \), conclude that the geometric series diverges by the geometric series test.

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Key Concepts

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

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether such a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of series like the one given.
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Convergence of an Infinite Series

Geometric Series

A geometric series has terms that multiply by a constant ratio each time. It converges if the absolute value of this ratio is less than one, and its sum can be found using a specific formula. Recognizing the given series as geometric simplifies the convergence test.
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Geometric Series

Convergence Tests

Convergence tests, such as the Ratio Test or Root Test, help determine if a series converges or diverges. These tests analyze the limit of term ratios or roots to conclude about convergence. Choosing an appropriate test is key to solving series problems efficiently.
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Choosing a Convergence Test
Related Practice
Textbook Question

Finding steady states using infinite series Solve Exercise 40 by expressing the amount of aspirin in your blood as a geometric series and evaluating the series.

Textbook Question

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 0 to ∞)(tan⁻¹(k + 2) − tan⁻¹k)

Textbook Question

{Use of Tech} Error in a finite alternating sum

How many terms of the series ∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k⁴ must be summed to ensure that the approximation error is less than 10⁻⁸?

Textbook Question

25–26. Recursively defined sequences

The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.


a.Find the first five terms a₀, a₁, ..., a₄ of each sequence.


25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

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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.

∑ (from k = 1 to ∞)2ᵏ / eᵏ

Textbook Question

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

∑ (from k = 1 to ∞)(7 + sin k) / k²