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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 10.2.47
Capitolo 10, Problema 10.2.47

13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.


{(1 / n)¹⁄ⁿ}

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Identify the given sequence: \(a_n = \left( \frac{1}{n} \right)^{\frac{1}{n}}\).
Rewrite the sequence using properties of exponents and logarithms to make the limit easier to analyze: \(a_n = e^{\ln \left( \left( \frac{1}{n} \right)^{\frac{1}{n}} \right)} = e^{\frac{1}{n} \ln \left( \frac{1}{n} \right)}\).
Simplify the exponent: \(\frac{1}{n} \ln \left( \frac{1}{n} \right) = \frac{1}{n} (-\ln n) = -\frac{\ln n}{n}\).
Analyze the limit of the exponent as \(n\) approaches infinity: \(\lim_{n \to \infty} -\frac{\ln n}{n}\). Recall that \(\ln n\) grows slower than \(n\), so this limit tends to zero.
Conclude the limit of the sequence by applying the limit to the exponential form: \(\lim_{n \to \infty} a_n = e^{\lim_{n \to \infty} -\frac{\ln n}{n}} = e^0\).

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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 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

Properties of Exponents and Roots

Understanding how to manipulate expressions with fractional exponents and roots is essential. For example, (1/n)^(1/n) can be rewritten using exponent rules to analyze its behavior as n increases.
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Percorso guidato
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Properties of Functions

Use of Logarithms in Limit Evaluation

Taking the natural logarithm of a sequence can simplify limit evaluation, especially for expressions involving exponents. By analyzing the limit of the logarithm, one can often find the original sequence's limit using continuity of the exponential function.
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Evaluate Logarithms
Pratica correlata
Domanda del libro di testo

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) 3ᵏ / (k² + 1)

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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 ∞) 1 / (k² + 4)

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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.


67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)

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Domanda del libro di testo

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.


{1.00001ⁿ}

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Domanda del libro di testo

Compare the growth rates of {n¹⁰⁰} and {eⁿ⁄¹⁰⁰} as n → ∞.

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Domanda del libro di testo

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

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