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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 10.6.23
Capitolo 10, Problema 10.6.23

11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ (k¹¹ + 2k⁵ + 1) / [4k(k¹⁰ + 1)]

Guida verificata passo dopo passo
1
Identify the general term of the series: \(a_k = ( -1 )^k \frac{k^{11} + 2k^5 + 1}{4k(k^{10} + 1)}\). Notice the factor \((-1)^k\) indicates this is an alternating series.
To apply the Alternating Series Test, focus on the absolute value of the terms without the alternating sign: \(b_k = \frac{k^{11} + 2k^5 + 1}{4k(k^{10} + 1)}\).
Simplify \(b_k\) by dividing numerator and denominator by the highest power of \(k\) in the denominator, which is \(k^{11}\), to analyze the limit as \(k \to \infty\).
Check the limit \(\lim_{k \to \infty} b_k\). If this limit is zero, proceed to the next step; otherwise, the series does not converge by the Alternating Series Test.
Verify if the sequence \(b_k\) is eventually decreasing for sufficiently large \(k\). If \(b_k\) is decreasing and the limit is zero, then by the Alternating Series Test, the series converges.

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Alternating Series Test

The Alternating Series Test determines the convergence of series whose terms alternate in sign. It requires that the absolute value of the terms decreases monotonically to zero. If these conditions hold, the series converges, even if it does not converge absolutely.
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Percorso guidato
10:54
Alternating Series Test

Behavior of the General Term

Analyzing the general term's behavior as k approaches infinity is crucial. Simplifying the term helps identify its limit and whether it approaches zero, a necessary condition for convergence of any infinite series.
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05:44
Divergence Test (nth Term Test)

Comparison and Simplification of Polynomial Expressions

Simplifying complex polynomial expressions in the numerator and denominator helps understand the dominant terms. This simplification aids in evaluating limits and determining the term's asymptotic behavior, which is essential for applying convergence tests.
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Taylor Polynomials
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Domanda del libro di testo

35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 

aₙ = 3 + cos(π*ⁿ) ; n = 1, 2, 3, …

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Domanda del libro di testo

Limits of sequences

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


{nsin(6 / n)}

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Does a geometric series always have a finite value?

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


25.∑ (k = 0 to ∞) 0.9ᵏ

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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 ∞) 1 / ∛k

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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.


{nsin³(nπ / 2) / (n + 1)}"

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