Domanda del libro di testo
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(n + 1)!⁄n!}
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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 10.2.35
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹(n)}
Guida verificata passo dopo passo1
Recognize that the sequence is given by \( a_n = \tan^{-1}(n) \), where \( \tan^{-1} \) is the inverse tangent function, also known as arctangent.
Recall the behavior of the arctangent function: as its input \( x \) approaches infinity, \( \tan^{-1}(x) \) approaches a horizontal asymptote at \( \frac{\pi}{2} \).
Since \( n \) is a sequence that increases without bound (\( n \to \infty \)), consider the limit \( \lim_{n \to \infty} \tan^{-1}(n) \).
Use the known limit property: \( \lim_{x \to \infty} \tan^{-1}(x) = \frac{\pi}{2} \). This implies the sequence \( a_n = \tan^{-1}(n) \) converges to \( \frac{\pi}{2} \).
Conclude that the limit of the sequence \( \{ \tan^{-1}(n) \} \) as \( n \to \infty \) is \( \frac{\pi}{2} \).
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Limits of Sequences
The limit of a sequence is the value that the terms of the sequence approach as the index 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
Inverse Tangent Function (arctan)
The inverse tangent function, arctan(x), returns the angle whose tangent is x. It is defined for all real numbers and has horizontal asymptotes at ±π/2, meaning as x approaches ±∞, arctan(x) approaches ±π/2.
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Behavior of arctan(n) as n → ∞
As the input n grows without bound, arctan(n) approaches its horizontal asymptote π/2. This means the sequence {arctan(n)} converges to π/2, since the values get closer and closer to this limit for large n.
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5:52Writing a General Formula Example 1
Pratica correlata
Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞)3k / ∜(k⁴ + 3)
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Domanda del libro di testo
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ · (k!) / (kᵏ)(Hint: Show that k! / kᵏ ≤ 2 / k², for k ≥ 3.)
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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 / (k + 10))ᵏ
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Domanda del libro di testo
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (−1)ᵏ / k^(2/3)
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Domanda del libro di testo
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{n³⁄(n⁴ + 1)}
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