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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 10.8.69
Capitolo 10, Problema 10.8.69

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


∑ (from k = 1 to ∞)(1 + 1 / (2k))ᵏ

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First, identify the general term of the series: \(a_k = \left(1 + \frac{1}{2k}\right)^k\).
To determine convergence, consider the behavior of \(a_k\) as \(k\) approaches infinity. Calculate the limit \(\lim_{k \to \infty} a_k = \lim_{k \to \infty} \left(1 + \frac{1}{2k}\right)^k\).
Recognize that this limit resembles the form of the exponential function \(e^x\), where \(\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x\). Here, \(x = \frac{1}{2}\), so the limit becomes \(e^{1/2}\).
Since the limit of the terms \(a_k\) is \(e^{1/2}\), which is a positive number not equal to zero, apply the Divergence Test (also called the Test for Divergence), which states that if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges.
Conclude that because the terms do not approach zero, the series \(\sum_{k=1}^\infty \left(1 + \frac{1}{2k}\right)^k\) diverges.

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Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.
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Convergence of an Infinite Series

Root Test for Convergence

The root test evaluates the limit of the k-th root of the absolute value of the terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. This test is particularly useful for series with terms raised to the k-th power.
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Root Test

Behavior of Terms in the Series

Analyzing the general term (1 + 1/(2k))^k involves understanding limits and growth rates. Recognizing how the term behaves as k approaches infinity helps in applying convergence tests and deciding if terms approach zero, a necessary condition for convergence.
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Geometric Series
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11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)

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45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 1 to ∞) cos(k) / k³

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54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


59. ∑ (k = –3 to ∞) 4 / ((4k – 3)(4k + 1))

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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.


ln 2 = ∑ (k = 1 to ∞) (−1)ᵏ⁺¹ / k

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55–70. More sequences

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


{cosn / n}

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

∑ (from k = 1 to ∞) 2ᵏ / (3ᵏ − 2ᵏ)

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