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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.R.93
Capitolo 10, Problema 10.R.93

{Use of Tech} Error in a finite alternating sum
How many terms of the series ∑ (from k = 1 to ∞)(−1)ᵏ⁺¹ / k⁴ must be summed to ensure that the approximation error is less than 10⁻⁸?

Guida verificata passo dopo passo
1
Recognize that the given series is an alternating series of the form \( \sum_{k=1}^\infty (-1)^{k+1} \frac{1}{k^4} \), which satisfies the conditions of the Alternating Series Test: the terms \( \frac{1}{k^4} \) are positive, decreasing, and approach zero as \( k \to \infty \).
Recall the Alternating Series Estimation Theorem, which states that the absolute error when approximating the sum by the first \( n \) terms is less than or equal to the absolute value of the first omitted term, i.e., \( |R_n| \leq \left| a_{n+1} \right| \).
Set up the inequality for the error bound: \( \left| a_{n+1} \right| = \frac{1}{(n+1)^4} < 10^{-8} \), since the error must be less than \( 10^{-8} \).
Solve the inequality \( \frac{1}{(n+1)^4} < 10^{-8} \) by taking reciprocals and then the fourth root: \( (n+1)^4 > 10^{8} \) which implies \( n+1 > (10^{8})^{1/4} \).
Calculate \( (10^{8})^{1/4} = 10^{8/4} = 10^{2} = 100 \), so \( n+1 > 100 \) and therefore \( n \geq 100 \). This means you must sum at least 100 terms to ensure the error is less than \( 10^{-8} \).

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Alternating Series and Alternating Series Test

An alternating series is a series whose terms alternate in sign. The Alternating Series Test states that if the absolute value of the terms decreases monotonically to zero, the series converges. This test also provides a way to estimate the error when approximating the sum by partial sums.
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Alternating Series Test

Error Bound for Alternating Series

For an alternating series that meets the conditions of the Alternating Series Test, the error made by stopping at the nth term is less than or equal to the absolute value of the (n+1)th term. This allows us to control the approximation error by choosing a sufficient number of terms.
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Geometric Series

Convergence Rate of p-Series with p=4

The given series involves terms of the form 1/k⁴, which is a p-series with p=4. Since p > 1, the series converges rapidly. Understanding the rate of decay of terms helps in estimating how many terms are needed to achieve a desired accuracy.
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P-Series and Harmonic Series
Pratica correlata
Domanda del libro di testo

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (–1)ⁿ / 0.9ⁿ

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Domanda del libro di testo

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.


∑ (k = 0 to ∞) k / (2k + 1)

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Domanda del libro di testo

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

∑ (from k = 1 to ∞)sin(1 / k⁹)

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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.

∑ (k = 1 to ∞) k / √(k² + 1)

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Domanda del libro di testo

25–26. Recursively defined sequences

The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.


b.Determine the limit of each sequence.


25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

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Domanda del libro di testo

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−1)ᵏ⁺¹(k² + 4) / (2k² + 1)

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