Domanda del libro di testo
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (−1)ᵏ / √(k³ᐟ² + k)
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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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 10, Problema 10.6.29
8–32. {Use of Tech} Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S − Sₙ| in using the nth partial sum Sₙ to estimate the value of the series S.
∑ (k = 1 to ∞) (−1)ᵏ / k⁴; n = 4
Guida verificata passo dopo passo1
Identify the given alternating series: \( \sum_{k=1}^{\infty} \frac{(-1)^k}{k^4} \). This is an alternating series where the terms are \( a_k = \frac{1}{k^4} \).
Calculate the 4th partial sum \( S_4 \) by summing the first 4 terms of the series: \( S_4 = \sum_{k=1}^4 \frac{(-1)^k}{k^4} = \frac{(-1)^1}{1^4} + \frac{(-1)^2}{2^4} + \frac{(-1)^3}{3^4} + \frac{(-1)^4}{4^4} \).
Recall Theorem 10.18 (Alternating Series Estimation Theorem), which states that the absolute error \( |S - S_n| \) when approximating the sum \( S \) by the partial sum \( S_n \) is less than or equal to the absolute value of the first omitted term: \( |S - S_n| \leq |a_{n+1}| \).
Find the upper bound for the error by evaluating the absolute value of the next term after the 4th partial sum: \( |a_5| = \left| \frac{(-1)^5}{5^4} \right| = \frac{1}{5^4} \).
Summarize the results: the 4th partial sum \( S_4 \) approximates the series sum \( S \), and the error in this approximation is at most \( \frac{1}{5^4} \).
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Alternating Series
An alternating series is a series whose terms alternate in sign, typically expressed as (-1)^k times a positive term. Such series often converge under specific conditions, especially when the absolute value of terms decreases monotonically to zero. Understanding the structure of alternating series is crucial for applying convergence tests and error estimates.
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06:00
Geometric Series
Partial Sums of a Series
A partial sum Sₙ is the sum of the first n terms of a series and serves as an approximation to the series' total sum S. Calculating partial sums helps estimate the value of infinite series by truncating after a finite number of terms, which is essential for practical computations and error analysis.
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06:45Intro to Series: Partial Sums
Error Bound for Alternating Series (Theorem 10.18)
Theorem 10.18 states that for a convergent alternating series with decreasing terms, the absolute error |S − Sₙ| in approximating the sum by the nth partial sum is at most the absolute value of the (n+1)th term. This provides a straightforward way to estimate the maximum error when using partial sums.
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10:54Alternating Series Test
Pratica correlata
Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
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Domanda del libro di testo
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 20 / (∛k + √k)
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Domanda del libro di testo
1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 1 to ∞) (k² / (k⁴ + k³ + 1))
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Domanda del libro di testo
54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.
63. ∑ (k = 1 to ∞) 1 / ((k + p)(k + p + 1)), where p is a positive integer
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Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
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