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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 10.8.57
Capitolo 10, Problema 10.8.57

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)5¹⁻²ᵏ

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Identify the given series: \( \sum_{k=1}^{\infty} 5^{1 - 2k} \). Notice that the general term can be rewritten to better understand its form.
Rewrite the term \( 5^{1 - 2k} \) as \( 5^{1} \cdot 5^{-2k} = 5 \cdot (5^{-2})^{k} \). This shows the series is a geometric series with common ratio \( r = 5^{-2} = \frac{1}{25} \).
Recall the convergence criterion for a geometric series: it converges if and only if \( |r| < 1 \). Since \( \frac{1}{25} < 1 \), the series converges.
To justify convergence, state that because the absolute value of the common ratio is less than 1, the infinite sum converges to \( \frac{a}{1 - r} \), where \( a \) is the first term of the series.
Find the first term \( a = 5^{1 - 2(1)} = 5^{-1} = \frac{1}{5} \) and express the sum formula for the series as \( S = \frac{a}{1 - r} = \frac{\frac{1}{5}}{1 - \frac{1}{25}} \), which confirms the series converges and allows calculation of its sum.

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Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, and its convergence depends on the absolute value of the ratio r. If |r| < 1, the series converges; otherwise, it diverges.
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Geometric Series

Convergence Tests

Convergence tests are methods used to determine whether an infinite series converges or diverges. For geometric series, the ratio test or direct recognition of the ratio can be applied. Understanding these tests helps justify the behavior of the series.
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Choosing a Convergence Test

Exponential Expressions and Simplification

Simplifying exponential expressions is crucial to identify the common ratio in a series. Rewriting terms like 5^(1-2k) helps express the series in a standard geometric form, making it easier to analyze convergence.
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Simplifying Exponential Expressions
Pratica correlata
Domanda del libro di testo

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 5ᵏ(k!)² / (2k)!

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Domanda del libro di testo

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.


aₙ = e⁻ⁿcosn

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Domanda del libro di testo

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


∑ (k = 1 to ∞) 4ᵏ / (5ᵏ − 3)

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Domanda del libro di testo

17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.

∑ (k = 1 to ∞) 1 / (∛(5k + 3))

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Domanda del libro di testo

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


∑ (k = 1 to ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))

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Domanda del libro di testo

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


33.∑ (k = 4 to ∞) 1 / 5ᵏ

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