Domanda del libro di testo
21–42. Geometric series Evaluate each geometric series or state that it diverges.
39.∑ (k = 2 to ∞) (–0.15)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
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Capitolo 10, Problema 10.2.51
Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{nsin(6 / n)}
Guida verificata passo dopo passo1
Identify the sequence given: \(a_n = n \sin\left(\frac{6}{n}\right)\).
Recall the limit property for sine near zero: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).
Rewrite the sequence term to use this property by setting \(x = \frac{6}{n}\), so \(a_n = n \sin\left(\frac{6}{n}\right) = \frac{\sin\left(\frac{6}{n}\right)}{\frac{6}{n}} \times 6\).
Analyze the limit as \(n \to \infty\): since \(\frac{6}{n} \to 0\), use the limit property to find \(\lim_{n \to \infty} \frac{\sin\left(\frac{6}{n}\right)}{\frac{6}{n}} = 1\).
Combine the results to conclude that \(\lim_{n \to \infty} a_n = 6 \times 1 = 6\).
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Limit of a Sequence
The limit of a sequence is the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.
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Introduction to Sequences
Behavior of the Sine Function for Small Arguments
For values of x close to zero, sin(x) is approximately equal to x. This linear approximation, sin(x) ≈ x, is useful for evaluating limits involving sine functions where the argument tends to zero.
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Percorso guidato
5:53Graph of Sine and Cosine Function
Limit Laws and Substitution
Limit laws allow the evaluation of limits by breaking complex expressions into simpler parts. Substitution involves replacing variables with their limiting values when the function is continuous, facilitating the calculation of the sequence's limit.
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05:21Finding Limits by Direct Substitution
Pratica correlata
Domanda del libro di testo
9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.
∑ (from k = 1 to ∞) (2ᵏ / k⁹⁹)
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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
Does a geometric series always have a finite value?
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Domanda del libro di testo
21–42. Geometric series Evaluate each geometric series or state that it diverges.
25.∑ (k = 0 to ∞) 0.9ᵏ
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Domanda del libro di testo
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ (k¹¹ + 2k⁵ + 1) / [4k(k¹⁰ + 1)]
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