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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.3.87b
Capitolo 10, Problema 10.3.87b

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.

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Understand the problem: We are asked to determine if the statement "If \( \sum_{k=1}^{\infty} a_k \) diverges, then \( \sum_{k=10}^{\infty} a_k \) also diverges" is true or false.
Recall the definition of series convergence and divergence: A series \( \sum_{k=m}^{\infty} a_k \) converges if the sequence of partial sums \( S_n = \sum_{k=m}^n a_k \) approaches a finite limit as \( n \to \infty \). Otherwise, it diverges.
Analyze the relationship between the two series: The series starting at \( k=10 \) is essentially the tail of the series starting at \( k=1 \). The original series can be written as \( \sum_{k=1}^{\infty} a_k = \sum_{k=1}^{9} a_k + \sum_{k=10}^{\infty} a_k \).
Consider the impact of the finite sum \( \sum_{k=1}^{9} a_k \): Since this is a finite sum, it does not affect convergence or divergence of the infinite series. Therefore, the convergence or divergence of \( \sum_{k=10}^{\infty} a_k \) determines the behavior of the tail.
Conclude based on the above: If the entire series \( \sum_{k=1}^{\infty} a_k \) diverges, then its tail \( \sum_{k=10}^{\infty} a_k \) must also diverge, because adding or removing a finite number of terms does not change the divergence property.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms from a sequence. A series converges if its partial sums approach a finite limit; otherwise, it diverges. Understanding convergence is essential to analyze whether changing the starting index affects the series' behavior.
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Convergence of an Infinite Series

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Shifting the starting index of a series changes only a finite number of terms. Since convergence depends on the tail behavior of the series, adding or removing finitely many terms does not affect whether the series converges or diverges.
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Counterexamples in Series Analysis

Counterexamples demonstrate when a general statement is false. To test if a series starting at k=10 diverges when the series from k=1 diverges, one can consider series where initial terms affect convergence, highlighting the importance of examining specific cases.
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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

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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.

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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, ......}

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Suppose the sequence {aₙ}⁽∞⁾ₙ₌₀ is defined by the recurrence relation

aₙ₊₁ = ⅓aₙ + 6;a₀ = 3.


b.Explain why {aₙ}⁽∞⁾ₙ₌₀ converges and find the limit.

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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

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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

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