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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.R.67
Capitolo 10, Problema 10.R.67

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k⁵ e⁻ᵏ

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Identify the series given: \( \sum_{k=1}^{\infty} k^{5} e^{-k} \). This is an infinite series where the general term is \( a_k = k^{5} e^{-k} \).
Recognize that the term \( e^{-k} \) can be rewritten as \( \left( \frac{1}{e} \right)^k \), which is an exponential decay factor.
Consider using the Ratio Test for convergence, which is effective for series involving factorials, exponentials, or powers. The Ratio Test states to compute \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \).
Calculate the ratio \( \frac{a_{k+1}}{a_k} = \frac{(k+1)^5 e^{-(k+1)}}{k^5 e^{-k}} = \frac{(k+1)^5}{k^5} \cdot e^{-1} \). Simplify this expression to prepare for taking the limit as \( k \to \infty \).
Evaluate the limit \( L = \lim_{k \to \infty} \frac{(k+1)^5}{k^5} \cdot e^{-1} \). Since \( \frac{(k+1)^5}{k^5} \to 1 \), the limit simplifies to \( L = e^{-1} \). Use the Ratio Test conclusion: if \( L < 1 \), the series converges; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive.

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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 ∑ k⁵ e⁻ᵏ.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests help determine convergence by comparing the given series to a known benchmark series. The Comparison Test checks if terms are smaller than those of a convergent series, while the Limit Comparison Test uses the limit of the ratio of terms. They are useful when terms involve products like polynomial and exponential functions.
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Limit Comparison Test

Behavior of Exponential vs. Polynomial Functions

Exponential functions like e⁻ᵏ decay faster than any polynomial grows as k approaches infinity. This means terms like k⁵ e⁻ᵏ tend to zero rapidly, often ensuring convergence of the series. Recognizing this interplay helps in selecting appropriate convergence tests.
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Graphs of Exponential Functions
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Domanda del libro di testo

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

∑ (from k = 1 to ∞)k⁴ / √(9k¹² + 2)

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Domanda del libro di testo

Sequences versus series

a.Find the limit of the sequence { (−⁴⁄₅)ᵏ }.

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Domanda del libro di testo

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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Domanda del libro di testo

Explain why or why not

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


e.The sequence aₙ = n² / (n² + 1) converge.

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Domanda del libro di testo

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

∑ (from k = 1 to ∞)k√k / k³

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Domanda del libro di testo

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

∑ (from k = 1 to ∞)k! / (eᵏ kᵏ)

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