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

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.2.7

Capitolo 10, Problema 10.2.7

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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Identify the given sequence: $\{1.00001^n\}$, where $n$ is a positive integer increasing without bound.

Recall that for sequences of the form $a^n$, the behavior depends on the base $a$: if $|a| > 1$, the sequence grows without bound; if $|a| = 1$, it is constant or oscillates; if $|a| < 1$, it converges to zero.

Since $1.00001$ is slightly greater than 1, the sequence \$1.00001^n$ will increase as $n$ increases, tending towards infinity, so it does not converge to a finite limit.

Determine monotonicity: because the base is greater than 1 and positive, each term is larger than the previous one, so the sequence is strictly increasing and monotonic.

Summarize: the sequence diverges (does not converge to a finite limit) and is monotonic increasing; it does not oscillate.

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Sequence Convergence and Divergence

A sequence converges if its terms approach a specific finite limit as n approaches infinity. If the terms do not approach any finite value, the sequence diverges. Determining convergence involves analyzing the behavior of the general term for large n.

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Monotonicity of Sequences

A sequence is monotonic if it is either entirely non-increasing or non-decreasing. Monotonic sequences have terms that consistently move in one direction, which helps in understanding their long-term behavior and potential convergence.

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Introduction to Sequences

Exponential Sequences and Limits

Sequences of the form a^n, where a is a constant, exhibit different behaviors depending on the value of a. If |a| > 1, the sequence grows without bound (diverges); if |a| < 1, it converges to zero; if a = 1, it is constant. This concept is key to analyzing the given sequence 1.00001^n.

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Pratica correlata

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 = 3 to ∞) 1 / (k − 2)⁴

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

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