Skip to main content
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 10.2.69
Capitolo 10, Problema 10.2.69

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


{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

Guida verificata passo dopo passo
1
Identify the given sequence as \( a_n = \frac{75}{n \cdot 99^n} + \frac{5^n \sin n}{8^n} \). We want to find \( \lim_{n \to \infty} a_n \) or determine if it diverges.
Analyze the first term \( \frac{75}{n \cdot 99^n} \): since \( 99^n \) grows exponentially and \( n \) grows linearly, this term approaches zero as \( n \to \infty \).
Analyze the second term \( \frac{5^n \sin n}{8^n} \): rewrite it as \( \sin n \cdot \left( \frac{5}{8} \right)^n \). Since \( \left( \frac{5}{8} \right)^n \) is an exponential decay (because \( \frac{5}{8} < 1 \)) and \( \sin n \) is bounded between -1 and 1, this term also approaches zero as \( n \to \infty \).
Combine the limits of both terms: since both approach zero, the sum \( a_n \) approaches zero as \( n \to \infty \).
Conclude that the sequence converges and its limit is zero.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Limits of Sequences

The limit of a sequence describes the value that the terms approach as the index n goes to infinity. Understanding how to evaluate limits helps determine whether a sequence converges to a finite number or diverges. Techniques often involve analyzing the behavior of individual terms or applying limit laws.
Video consigliato:
8:22
Introduction to Sequences

Exponential Growth and Decay

Exponential terms like 99ⁿ, 5ⁿ, and 8ⁿ grow or decay rapidly depending on the base. Comparing bases helps determine which terms dominate as n increases. For example, if the denominator grows faster than the numerator, the fraction tends to zero, influencing the sequence's limit.
Video consigliato:
09:29
Exponential Growth & Decay

Behavior of Oscillatory Functions in Sequences

Functions like sin(n) oscillate between -1 and 1, causing terms to fluctuate. When multiplied by terms that tend to zero, the oscillations become negligible, often leading to convergence. Understanding how oscillatory factors interact with exponential terms is key to analyzing sequence limits.
Video consigliato:
8:22
Introduction to Sequences
Pratica correlata
Domanda del libro di testo

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)

53
views
Domanda del libro di testo

13–52. Limits of sequences

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


{√((1 + 1 / 2n)ⁿ)}

37
views
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 / (2√k − 1)

94
views
Domanda del libro di testo

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.


π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

61
views
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 / (2k − √k)

81
views
Domanda del libro di testo

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-1)ᵏ⁺¹ × k²ᵏ) / (k! × k!)

58
views