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 10.7.13

Capitolo 10, Problema 10.7.13

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) (k² / 4ᵏ)

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Identify the series given: \( \sum_{k=1}^{\infty} \frac{k^{2}}{4^{k}} \). We want to determine if it converges absolutely or diverges.

Recall the Ratio Test: For a series \( \sum a_k \), compute the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive.

Set \( a_k = \frac{k^{2}}{4^{k}} \). Compute \( \frac{a_{k+1}}{a_k} = \frac{(k+1)^{2} / 4^{k+1}}{k^{2} / 4^{k}} = \frac{(k+1)^{2}}{k^{2}} \cdot \frac{1}{4} \).

Take the limit as \( k \to \infty \) of \( \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \frac{(k+1)^{2}}{k^{2}} \cdot \frac{1}{4} = \left( \lim_{k \to \infty} \frac{(k+1)^{2}}{k^{2}} \right) \cdot \frac{1}{4} \).

Evaluate the limit \( \lim_{k \to \infty} \frac{(k+1)^{2}}{k^{2}} = 1 \), so the overall limit \( L = 1 \cdot \frac{1}{4} = \frac{1}{4} \). Since \( L < 1 \), by the Ratio Test, the series converges absolutely.

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Ratio Test

The Ratio Test determines the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.

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05:35

Ratio Test

Root Test

The Root Test analyzes the nth root of the absolute value of the nth term of a series. If the limit of this root is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. It is especially useful for series with terms raised to the nth power.

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07:15

Root Test

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence of the original series and is a stronger condition than conditional convergence, ensuring stability under rearrangement of terms.

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07:51

Choosing a Convergence Test

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23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k^(–1/5)

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9–15. Geometric sums Evaluate each geometric sum.

{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ

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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.

aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)

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