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 10.1.71b
Capitolo 10, Problema 10.1.71b

Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.


b.If a sequence of positive numbers converges, then the sequence is decreasing.

Guida verificata passo dopo passo
1
Recall the definition of a sequence converging: A sequence \( \{a_n\} \) converges to a limit \( L \) if for every \( \epsilon > 0 \), there exists an \( N \) such that for all \( n > N \), \( |a_n - L| < \epsilon \).
Understand what it means for a sequence to be decreasing: A sequence \( \{a_n\} \) is decreasing if \( a_{n+1} \leq a_n \) for all \( n \).
Analyze the statement: "If a sequence of positive numbers converges, then the sequence is decreasing." This implies that every convergent sequence of positive terms must be decreasing.
Consider a counterexample to test the statement: For instance, the sequence \( a_n = \frac{1}{n} \) is positive and converges to 0, and it is decreasing. However, the sequence \( a_n = \frac{1}{n} \) is decreasing, but what about a sequence like \( a_n = \frac{1}{n} \) for odd \( n \) and \( a_n = \frac{1}{n} + \frac{1}{n^2} \) for even \( n \)? This sequence is positive and converges to 0 but is not strictly decreasing.
Conclude that convergence of a positive sequence does not guarantee it is decreasing, so the statement is false. A convergent sequence can oscillate or increase at some points as long as it approaches the limit.

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.

Convergence of a Sequence

A sequence converges if its terms approach a specific finite limit as the index goes to infinity. Convergence does not impose restrictions on the sequence's monotonicity; the terms can oscillate or increase before settling near the limit.
Video consigliato:
8:22
Introduction to Sequences

Monotonic Sequences

A sequence is decreasing if each term is less than or equal to the previous term. Monotonicity is a property describing the sequence's order, independent of whether it converges or not.
Video consigliato:
8:22
Introduction to Sequences

Counterexamples in Sequence Analysis

To disprove a statement about sequences, providing a counterexample—a sequence that meets the conditions but violates the conclusion—is effective. For instance, a convergent sequence of positive numbers that is not decreasing shows the statement is false.
Video consigliato:
8:22
Introduction to Sequences
Pratica correlata
Domanda del libro di testo

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. (2n)! / (2n − 1)! = 2n

61
views
Domanda del libro di testo

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).


{1, 3, 9, 27, 81, ......}

52
views
Domanda del libro di testo

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


b. Find an explicit formula for the nth term of the sequence {hₙ}.


h₀ = 20,r = 0.5

82
views
Domanda del libro di testo

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

46
views
Domanda del libro di testo

18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.


b. Evaluate the series using Theorem 10.7.


∑ (k = 0 to ∞) (–2/7)ᵏ

79
views
Domanda del libro di testo

39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.


b. Find an upper bound for the remainder Rₙ.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

120
views