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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 10.5.41
Capitolo 10, Problema 10.5.41

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


∑ (k = 1 to ∞) (1 + 2 / k)ᵏ

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Identify the given series: \( \sum_{k=1}^{\infty} \left(1 + \frac{2}{k}\right)^k \). This is an infinite series where each term is \( a_k = \left(1 + \frac{2}{k}\right)^k \).
Recognize that the terms \( a_k \) resemble the form \( \left(1 + \frac{x}{k}\right)^k \), which is related to the expression for \( e^x \) as \( k \to \infty \). Here, \( x = 2 \).
Recall the Test for Divergence (also called the nth-term test): if \( \lim_{k \to \infty} a_k \neq 0 \), then the series \( \sum a_k \) diverges. So, compute \( \lim_{k \to \infty} \left(1 + \frac{2}{k}\right)^k \).
Evaluate the limit \( \lim_{k \to \infty} \left(1 + \frac{2}{k}\right)^k = e^2 \), which is a positive finite number greater than zero.
Since the limit of the terms \( a_k \) is not zero, by the Test for Divergence, the series \( \sum_{k=1}^{\infty} \left(1 + \frac{2}{k}\right)^k \) diverges.

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Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. Determining convergence involves analyzing the behavior of the terms and applying appropriate tests to see if the series sums to a finite value.
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Convergence of an Infinite Series

Root Test

The Root Test evaluates the limit of the k-th root of the absolute value of the terms in a series. 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 especially useful for series with terms raised to the k-th power.
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Limit Comparison and Growth of Terms

Understanding how the terms (1 + 2/k)^k behave as k approaches infinity is crucial. This expression resembles the form (1 + 1/n)^n, which approaches e, but with a different coefficient. Comparing the limit of terms to known limits helps determine if terms approach zero, a necessary condition for convergence.
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Limit Comparison Test
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Domanda del libro di testo

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k^(–1/5)

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9–15. Geometric sums Evaluate each geometric sum.


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Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.


aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)

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

∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)

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21–42. Geometric series Evaluate each geometric series or state that it diverges.  


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

∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)

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