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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 10.4.15
Capitolo 10, Problema 10.4.15

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.
∑ (k = 2 to ∞) √k / (ln¹⁰ k)

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Identify the general term of the series: \( a_k = \frac{\sqrt{k}}{(\ln k)^{10}} \) for \( k \geq 2 \).
Recall the Divergence Test (also known as the Test for Divergence): if \( \lim_{k \to \infty} a_k \neq 0 \), then the series \( \sum a_k \) diverges. If the limit equals zero, the test is inconclusive.
Calculate the limit \( \lim_{k \to \infty} \frac{\sqrt{k}}{(\ln k)^{10}} \). Consider the growth rates of the numerator and denominator: \( \sqrt{k} = k^{1/2} \) grows faster than any power of \( \ln k \).
Since \( k^{1/2} \) grows faster than \( (\ln k)^{10} \), the fraction \( \frac{\sqrt{k}}{(\ln k)^{10}} \) tends to infinity as \( k \to \infty \), so the limit does not equal zero.
Conclude that by the Divergence Test, because the limit of \( a_k \) is not zero, the series \( \sum_{k=2}^\infty \frac{\sqrt{k}}{(\ln k)^{10}} \) diverges.

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Divergence Test (Nth-Term Test)

The Divergence Test states that if the limit of the terms of a series does not approach zero as n approaches infinity, the series diverges. If the limit is zero, the test is inconclusive, and other methods must be used to determine convergence or divergence.
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05:44
Divergence Test (nth Term Test)

Behavior of the General Term

Analyzing the general term √k / (ln k)¹⁰ involves understanding how the numerator and denominator grow as k increases. Since √k grows faster than any power of ln k, the term's limit as k approaches infinity is crucial for applying the Divergence Test.
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Percorso guidato
05:44
Divergence Test (nth Term Test)

Logarithmic and Root Functions Growth Rates

Logarithmic functions grow slower than polynomial functions like square roots. Recognizing that √k increases faster than (ln k)¹⁰ helps determine the limit of the term, which is essential for deciding if the series terms approach zero or not.
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Graphs of Logarithmic Functions
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