Domanda del libro di testo
48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) 3ᵏ / (k² + 1)
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
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Capitolo 10, Problema 10.6.13
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)
Guida verificata passo dopo passo1
Identify the given series: \( \sum_{k=0}^{\infty} \frac{(-1)^k}{2k + 1} \). This is an alternating series because of the factor \( (-1)^k \), which causes the terms to alternate in sign.
Recall the Alternating Series Test (Leibniz Test), which states that an alternating series \( \sum (-1)^k a_k \) converges if two conditions are met: (1) the sequence \( a_k \) is positive, decreasing, and (2) \( \lim_{k \to \infty} a_k = 0 \).
Check the sequence \( a_k = \frac{1}{2k + 1} \): it is positive for all \( k \geq 0 \), and as \( k \) increases, \( 2k + 1 \) increases, so \( a_k \) decreases.
Evaluate the limit \( \lim_{k \to \infty} \frac{1}{2k + 1} \). Since the denominator grows without bound, the limit is 0.
Since both conditions of the Alternating Series Test are satisfied, conclude that the series \( \sum_{k=0}^{\infty} \frac{(-1)^k}{2k + 1} \) converges.
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Alternating Series Test
The Alternating Series Test determines if a series with terms alternating in sign converges. It requires that the absolute value of the terms decreases monotonically to zero. If these conditions hold, the series converges, though not necessarily absolutely.
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Percorso guidato
10:54
Alternating Series Test
Convergence of Infinite Series
An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence involves analyzing term behavior and applying tests like the Alternating Series Test or comparison tests to determine if the sum settles to a finite value.
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Percorso guidato
06:52Convergence of an Infinite Series
Behavior of the General Term
Examining the general term of a series, such as (−1)^k / (2k + 1), helps assess whether terms decrease in magnitude and approach zero. This behavior is crucial for applying convergence tests and ensuring the series meets necessary criteria for convergence.
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Percorso guidato
05:44Divergence Test (nth Term Test)
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Domanda del libro di testo
6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.
{1.00001ⁿ}
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Compare the growth rates of {n¹⁰⁰} and {eⁿ⁄¹⁰⁰} as n → ∞.
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Domanda del libro di testo
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(1 / n)¹⁄ⁿ}
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