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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.2.63
Capitolo 10, Problema 10.2.63

55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.


{tan⁻¹n / n}

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Identify the given sequence as \( a_n = \frac{\tan^{-1}(n)}{n} \), where \( \tan^{-1}(n) \) is the inverse tangent (arctangent) function.
Recall the behavior of the arctangent function as \( n \to \infty \): \( \tan^{-1}(n) \) approaches \( \frac{\pi}{2} \) because the arctangent of very large positive numbers tends to \( \frac{\pi}{2} \).
Rewrite the limit expression using this information: \( \lim_{n \to \infty} \frac{\tan^{-1}(n)}{n} = \lim_{n \to \infty} \frac{\frac{\pi}{2}}{n} \) approximately for large \( n \).
Since the denominator \( n \) grows without bound and the numerator approaches a constant \( \frac{\pi}{2} \), analyze the limit of a constant divided by an infinitely large number.
Conclude that the limit of the sequence is \( 0 \) because dividing a finite constant by an infinitely large number tends to zero.

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Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index goes to infinity. Understanding how to evaluate limits helps determine whether a sequence converges to a finite number or diverges.
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Introduction to Sequences

Behavior of the Arctangent Function

The arctangent function, tan⁻¹(x), is continuous and bounded, approaching π/2 as x approaches infinity. Knowing this asymptotic behavior is crucial for analyzing sequences involving tan⁻¹(n) as n grows large.
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Graphs of Exponential Functions

Growth Rates and Dominant Terms

Comparing the growth rates of numerator and denominator helps determine the limit of a sequence. Since n grows without bound and tan⁻¹(n) approaches a constant, understanding which term dominates is key to finding the sequence's limit.
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Percorso guidato
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Intro To Related Rates
Pratica correlata
Domanda del libro di testo

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{(3ⁿ⁺¹ + 3)⁄3ⁿ}

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11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)

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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.


{(1 + (2 / n))ⁿ}

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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.

∑ (from k = 3 to ∞) 5 / (2 + lnk)

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