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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 10.6.57
Capitolo 10, Problema 10.6.57

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ · k / (2k + 1)

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1
Identify the given series: \( \sum_{k=1}^{\infty} (-1)^k \cdot \frac{k}{2k + 1} \). This is an alternating series because of the factor \( (-1)^k \).
Check for absolute convergence by considering the absolute value of the terms: \( \sum_{k=1}^{\infty} \left| (-1)^k \cdot \frac{k}{2k + 1} \right| = \sum_{k=1}^{\infty} \frac{k}{2k + 1} \).
Analyze the behavior of the absolute value terms \( \frac{k}{2k + 1} \) as \( k \to \infty \). Determine if the series \( \sum \frac{k}{2k + 1} \) converges or diverges by comparing it to a simpler series.
If the absolute value series diverges, test the original alternating series for conditional convergence using the Alternating Series Test. Verify if the sequence \( b_k = \frac{k}{2k + 1} \) is decreasing and if \( \lim_{k \to \infty} b_k = 0 \).
Based on the results from the absolute convergence test and the Alternating Series Test, conclude whether the series converges absolutely, converges conditionally, or diverges.

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Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it implies the original series converges. Testing absolute convergence often involves comparison or ratio tests.
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Percorso guidato
07:51
Choosing a Convergence Test

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating series where the terms decrease in magnitude and approach zero.
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Percorso guidato
07:51
Choosing a Convergence Test

Alternating Series Test

The Alternating Series Test states that an alternating series ∑(−1)^k b_k converges if the sequence b_k is positive, decreasing, and approaches zero. This test helps determine conditional convergence when absolute convergence fails.
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Percorso guidato
10:54
Alternating Series Test
Pratica correlata
Domanda del libro di testo

55–70. More sequences

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


{sinn / 2ⁿ}

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Domanda del libro di testo

46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).

53.0.00952̅ = 0.00952952…

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Domanda del libro di testo

Growth rates of sequences

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


{n¹⁰⁰⁰ / 2ⁿ}

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Domanda del libro di testo

Is it possible for an alternating series to converge absolutely but not conditionally?

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Domanda del libro di testo

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 2 to ∞) (5lnk) / k

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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.

2 / 4² + 2 / 5² + 2 / 6² + ⋯

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