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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 10.2.55
Capitolo 10, Problema 10.2.55

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


{(−1)ⁿ / 2ⁿ}

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1
Identify the general term of the sequence, which is given by \(a_n = \frac{(-1)^n}{2^n}\).
Recall that \((-1)^n\) alternates between \(1\) and \(-1\) as \(n\) increases, causing the numerator to alternate signs.
Note that the denominator \$2^n$ grows exponentially as $n$ increases, becoming very large.
Consider the behavior of the absolute value of the terms: \(\left|a_n\right| = \frac{1}{2^n}\), which approaches \(0\) as \(n \to \infty\).
Since the numerator only changes sign but the magnitude approaches zero, conclude that the sequence converges to \(0\).

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Sequence and Limit

A sequence is an ordered list of numbers defined by a specific formula. The limit of a sequence is the value that the terms approach as the index goes to infinity. Understanding how to find limits helps determine if a sequence converges to a finite value or diverges.
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Introduction to Sequences

Behavior of Exponential Terms

Exponential terms like 2ⁿ grow rapidly as n increases. When in the denominator, such terms cause the overall fraction to approach zero. Recognizing this behavior is key to evaluating limits involving exponential expressions.
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Graphs of Exponential Functions

Alternating Sequences

An alternating sequence changes sign with each term, often represented by (−1)ⁿ. While the sign alternates, the magnitude may approach zero or another value. Understanding this helps analyze whether the sequence converges or oscillates without settling.
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Introduction to Sequences
Pratica correlata
Domanda del libro di testo

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (−1)ᵏ k (2ᵏ⁺¹ / (9ᵏ − 1))

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1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 1 to ∞) ((−1)ᵏ⁺¹) / (√2ᵏ + lnk)

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The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.

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38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.

39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 1)

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Domanda del libro di testo

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 2 to ∞) 1 / (klnk)

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Domanda del libro di testo

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) (−1)ᵏ × k / (k³ + 1)

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