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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 10.8.25
Chapter 10, Problem 10.8.25

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 1 / (√k × e^(√k))

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1
Identify the general term of the series: \(a_k = \frac{1}{\sqrt{k} \times e^{\sqrt{k}}}\).
Consider the behavior of the term \(a_k\) as \(k \to \infty\). Since \(e^{\sqrt{k}}\) grows very rapidly, the terms \(a_k\) approach zero.
To determine convergence, apply the Comparison Test by comparing \(a_k\) to a simpler series. Note that \(e^{\sqrt{k}}\) grows faster than any polynomial, so compare \(a_k\) to \(\frac{1}{e^{\sqrt{k}}}\).
Since \(\sum \frac{1}{e^{\sqrt{k}}}\) converges (because the terms decrease exponentially), and \(a_k < \frac{1}{e^{\sqrt{k}}}\) for all large \(k\), by the Comparison Test, the original series converges.
Alternatively, you can apply the Integral Test by considering the integral of the function \(f(x) = \frac{1}{\sqrt{x} e^{\sqrt{x}}}\) from 1 to infinity and showing that it converges, which implies the series converges.

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Key Concepts

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

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Determining convergence involves analyzing the behavior of the terms as the index grows large, ensuring the sum does not diverge to infinity.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests compare the given series to a known benchmark series. If terms of the given series behave similarly to a convergent or divergent series, we can conclude the same about the original series, simplifying convergence analysis.
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Limit Comparison Test

Exponential and Root Functions in Series Terms

Understanding how exponential functions like e^(√k) grow faster than polynomial or root functions is crucial. This growth rate often dominates the denominator, causing terms to decrease rapidly, which influences the convergence of the series.
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Exponential Functions
Related Practice
Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) (k² − 1) / (k³ + 4)

Textbook Question

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (−1)ᵏ k (2ᵏ⁺¹ / (9ᵏ − 1))

Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 1 to ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + lnk)

Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


41.∑ (k = 1 to ∞) 4 / 12ᵏ

Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)

Textbook Question

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (√k / k − 1)²ᵏ