Domanda del libro di testo
Suppose the sequence { aₙ} is defined by the recurrence relation a₍ₙ₊₁₎ = n · aₙ , for n=1, 2, 3 ...., where a₁ = 1. Write out the first five terms of the sequence.
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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 10.5.15
9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.
∑ (k = 1 to ∞) 4ᵏ / (5ᵏ − 3)
Guida verificata passo dopo passo1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{4^{k}}{5^{k} - 3} \). We want to determine if this series converges using the Comparison Test or the Limit Comparison Test.
Consider the behavior of the terms for large \( k \). Since \( 5^{k} \) grows much faster than 3, the denominator \( 5^{k} - 3 \) behaves approximately like \( 5^{k} \) for large \( k \). So the terms behave like \( \frac{4^{k}}{5^{k}} \).
Simplify the approximate terms: \( \frac{4^{k}}{5^{k}} = \left( \frac{4}{5} \right)^{k} \). This is a geometric series with ratio \( r = \frac{4}{5} \), which is less than 1, so the geometric series \( \sum \left( \frac{4}{5} \right)^{k} \) converges.
Use the Limit Comparison Test by computing \( \lim_{k \to \infty} \frac{\frac{4^{k}}{5^{k} - 3}}{\left( \frac{4}{5} \right)^{k}} \). Simplify this limit to check if it is a finite positive number.
If the limit is finite and positive, then by the Limit Comparison Test, the original series converges or diverges together with the geometric series \( \sum \left( \frac{4}{5} \right)^{k} \). Since the geometric series converges, conclude that the original series converges.
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Comparison Test
The Comparison Test determines the convergence of a series by comparing it to another series with known behavior. If the terms of the given series are less than or equal to the terms of a convergent series, then the given series also converges. Conversely, if the terms are greater than or equal to those of a divergent series, the given series diverges.
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Percorso guidato
09:25
Direct Comparison Test
Limit Comparison Test
The Limit Comparison Test involves taking the limit of the ratio of the terms of two series. If this limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the series have similar dominant behavior for large terms.
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Percorso guidato
07:45Limit Comparison Test
Geometric Series and Dominant Term Analysis
Understanding geometric series is essential since the given series involves terms like 4^k and 5^k. For large k, the dominant terms in numerator and denominator determine the behavior of the series. Recognizing that 4^k / 5^k behaves like (4/5)^k helps in comparing the series to a convergent geometric series.
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Percorso guidato
06:00Geometric Series
Pratica correlata
Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 5ᵏ(k!)² / (2k)!
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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.
aₙ = e⁻ⁿcosn
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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 ∞)5¹⁻²ᵏ
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Domanda del libro di testo
17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))
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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 ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))
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