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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.5.6
Capitolo 10, Problema 10.5.6

Using the Ratio Test
In Exercises 1–8, use the Ratio Test to determine whether each series converges absolutely or diverges.
∑(from n=2 to ∞) [(3ⁿ⁺²) / ln(n)]

Guida verificata passo dopo passo
1
Identify the general term of the series as \(a_n = \frac{3^{n+2}}{\ln(n)}\) for \(n \geq 2\).
Write the Ratio Test limit expression: compute \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
Substitute \(a_{n+1}\) and \(a_n\) into the ratio: \(\frac{a_{n+1}}{a_n} = \frac{\frac{3^{(n+1)+2}}{\ln(n+1)}}{\frac{3^{n+2}}{\ln(n)}} = \frac{3^{n+3}}{\ln(n+1)} \cdot \frac{\ln(n)}{3^{n+2}}\).
Simplify the ratio expression: \(\frac{a_{n+1}}{a_n} = 3 \cdot \frac{\ln(n)}{\ln(n+1)}\).
Evaluate the limit \(L = \lim_{n \to \infty} 3 \cdot \frac{\ln(n)}{\ln(n+1)}\) and use the Ratio Test criteria: if \(L < 1\), the series converges absolutely; if \(L > 1\), the series diverges; if \(L = 1\), the test is inconclusive.

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Ratio Test

The Ratio Test is a method to determine the convergence or divergence of an infinite series by examining the limit of the absolute value of the ratio of consecutive 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.
Video consigliato:
Percorso guidato
05:35
Ratio Test

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence of the original series and is a stronger condition than conditional convergence, simplifying the analysis of series with positive and negative terms.
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Percorso guidato
07:51
Choosing a Convergence Test

Behavior of Logarithmic and Exponential Functions in Series

Understanding how logarithmic functions grow slowly compared to exponential functions is crucial when analyzing series terms. In this series, the numerator grows exponentially while the denominator grows logarithmically, affecting the limit in the Ratio Test and the overall convergence behavior.
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06:32
Derivatives of General Logarithmic Functions
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Error Estimates

The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.

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Convergence or Divergence

Which of the series in Exercises 57–64 converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ (n!)ⁿ] / [n^(n²)]

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Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) 1 / (2√n + ³√n)

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Make up a geometric series ∑a rⁿ⁻¹ that converges to the number 5 if

b. a = 13/2

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Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (xⁿ / (2n + 1))^(1/n),x > 0

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In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.

∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ

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