Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
1 / (2·3) + 1 / (4·5) + 1 / (6·7) + 1 / (8·9) + ⋯
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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 10.7.35
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 ∞) (2k + 1)! / (k!)²
Guida verificata passo dopo passo1
First, identify the general term of the series: \(a_k = \frac{(2k + 1)!}{(k!)^2}\).
To determine convergence, consider using the Ratio Test, which is effective for factorial expressions. The Ratio Test states to compute \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\).
Write out the ratio \(\frac{a_{k+1}}{a_k} = \frac{(2(k+1) + 1)!}{((k+1)!)^2} \times \frac{(k!)^2}{(2k + 1)!}\) and simplify the factorial expressions carefully.
Evaluate the limit \(L\) as \(k\) approaches infinity. If \(L < 1\), the series converges absolutely; if \(L > 1\), the series diverges; if \(L = 1\), the test is inconclusive.
Based on the result of the Ratio Test, conclude whether the series converges absolutely, converges conditionally, or diverges.
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Absolute and Conditional Convergence
A series converges absolutely if the series of absolute values converges. If the original series converges but not absolutely, it converges conditionally. Understanding these distinctions helps classify the behavior of infinite series.
Video consigliato:
Percorso guidato
07:51
Choosing a Convergence Test
Ratio Test
The Ratio Test determines convergence by examining the limit of the ratio of consecutive terms. If the limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
Video consigliato:
Percorso guidato
05:35Ratio Test
Factorials and Growth Rates
Factorials grow very rapidly, often dominating polynomial terms. Comparing factorial expressions using simplification or the Ratio Test helps analyze the behavior of series terms and determine convergence or divergence.
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5:22Factorials
Pratica correlata
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13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (−1)ⁿ / 2ⁿ
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Domanda del libro di testo
21–42. Geometric series Evaluate each geometric series or state that it diverges.
37.1 + e/π + e²/π² + e³/π³ + ⋯
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46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
47.0.3̅ = 0.333…
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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 = 3 to ∞) 1 / lnk
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Domanda del libro di testo
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 ∞) k^(1/k)
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