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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 10.4.9
Capitolo 10, Problema 10.4.9

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.


∑ (k = 0 to ∞) k / (2k + 1)

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Identify the general term of the series: \(a_k = \frac{k}{2k + 1}\).
Recall the Divergence Test (also known as the nth-term test for divergence): if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges.
Calculate the limit of the general term as \(k\) approaches infinity: \(\lim_{k \to \infty} \frac{k}{2k + 1}\).
To find this limit, divide numerator and denominator by \(k\): \(\lim_{k \to \infty} \frac{1}{2 + \frac{1}{k}}\).
Evaluate the limit: as \(k \to \infty\), \(\frac{1}{k} \to 0\), so the limit becomes \(\frac{1}{2}\). Since this limit is not zero, by the Divergence Test, the series diverges.

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Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. If the limit is zero, the test is inconclusive, and other methods must be used to determine convergence.
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Divergence Test (nth Term Test)

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index k becomes very large. Evaluating this limit helps determine the behavior of the series' terms, which is essential for applying the Divergence Test.
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Introduction to Sequences

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Understanding whether such a series converges (approaches a finite value) or diverges (does not approach a finite value) is fundamental in calculus, often requiring tests like the Divergence Test.
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Convergence of an Infinite Series
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