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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 10.2.49
Capitolo 10, Problema 10.2.49

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


{bₙ}, where
bₙ = { n / (n + 1)if n ≤ 5000
ne⁻ⁿif n > 5000 }

Guida verificata passo dopo passo
1
Understand the definition of the sequence \( b_n \): it has two different expressions depending on whether \( n \) is less than or equal to 5000 or greater than 5000.
Analyze the first part of the sequence for \( n \leq 5000 \): \( b_n = \frac{n}{n+1} \). Since this is a rational function, consider the limit as \( n \to \infty \) to understand its behavior.
Analyze the second part of the sequence for \( n > 5000 \): \( b_n = n e^{-n} \). Recognize that \( e^{-n} = \frac{1}{e^n} \) and consider the limit as \( n \to \infty \) to determine how this term behaves.
Since the sequence changes definition at \( n = 5000 \), check the limit of both parts as \( n \to \infty \) to see if they approach the same value or if the sequence diverges.
Conclude the overall limit of the sequence \( b_n \) by combining the results from both parts and considering the behavior for large \( n \).

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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

Piecewise-Defined Sequences

A piecewise-defined sequence has different formulas for different ranges of n. To find its limit, analyze the behavior of each piece separately, especially the part that applies as n approaches infinity.
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Piecewise Functions

Exponential Decay and Its Limit

Sequences involving terms like n·e⁻ⁿ combine polynomial growth with exponential decay. Since exponential decay dominates polynomial growth, such terms tend to zero as n approaches infinity.
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Exponential Growth & Decay
Pratica correlata
Domanda del libro di testo

17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.

∑ (k = 1 to ∞) 1 / (∛(5k + 3))

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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 ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))

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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⁺¹ k¹/ᵏ

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Domanda del libro di testo

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


29.∑ (k = 1 to ∞) e^(–2k)

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Domanda del libro di testo

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 0 to ∞) 1 / (1000 + k)

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Domanda del libro di testo

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


33.∑ (k = 4 to ∞) 1 / 5ᵏ

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