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

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.5.7

Capitolo 10, Problema 10.5.7

What comparison series would you use with the Limit Comparison Test to determine whether ∑ (k = 1 to ∞) (k² + k + 5) / (k³ + 3k + 1) converges?

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Identify the general term of the series: \(a_k = \frac{k^2 + k + 5}{k^3 + 3k + 1}\).

To apply the Limit Comparison Test, focus on the dominant terms in the numerator and denominator for large \(k\) to understand the behavior of \(a_k\) as \(k \to \infty\).

The dominant term in the numerator is \(k^2\) and in the denominator is \(k^3\), so the term behaves like \(\frac{k^2}{k^3} = \frac{1}{k}\) for large \(k\).

Choose the comparison series \(b_k = \frac{1}{k}\), which is a \(p\)-series with \(p=1\).

Use the Limit Comparison Test by computing \(\lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} \frac{\frac{k^2 + k + 5}{k^3 + 3k + 1}}{\frac{1}{k}}\) to determine if the original series converges or diverges.

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Limit Comparison Test

The Limit Comparison Test is used to determine the convergence or divergence of an infinite series by comparing it to a second series with known behavior. It involves taking the limit of the ratio of the terms of the two series. If the limit is a positive finite number, both series either converge or diverge together.

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Percorso guidato

07:45

Limit Comparison Test

Dominant Terms in Polynomials

When analyzing series with polynomial terms, the highest degree terms dominate the behavior for large values of the index. Simplifying the terms by focusing on these dominant terms helps identify a simpler comparison series that approximates the original series' behavior at infinity.

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07:00

Taylor Polynomials

p-Series and Their Convergence

A p-series is a series of the form ∑ 1/k^p, which converges if and only if p > 1. Recognizing that the simplified dominant term resembles a p-series allows us to use known convergence properties to determine the behavior of the original series.

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04:30

P-Series and Harmonic Series

Pratica correlata

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51–56. {Use of Tech} Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.

aₙ₊₁ = 4aₙ + 1 a₀ = 1

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84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.

a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b.Use analytical methods to find the limit of the sequence.

aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3

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49–50. Limits from graphs Consider the following sequences. Find the first four terms of the sequence .Based on part (a) and the figure, determine a plausible limit of the sequence.

aₙ = 2 + 2⁻ⁿ;n = 1, 2, 3, …

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

∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)

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Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.

{n¹⁰ / ln20n}

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

∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)

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