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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.8.69
Chapter 10, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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
Related Practice
Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k² / (k³ + 1)

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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.


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Textbook Question

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


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Textbook Question

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