Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problema 10.PE.30

Capitolo 10, Problema 10.PE.30

Determining Convergence of Series
Which of the series in Exercises 25–44 converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers.

∑ (from n = 2 to ∞) 1/[n(ln n)²]

Guida verificata passo dopo passo

Identify the given series: \( \sum_{n=2}^{\infty} \frac{1}{n (\ln n)^2} \). We want to determine if it converges absolutely, conditionally, or diverges.

Since all terms \( \frac{1}{n (\ln n)^2} \) are positive for \( n \geq 2 \), absolute convergence is equivalent to convergence of the series itself.

Use the Integral Test to analyze convergence because the function \( f(x) = \frac{1}{x (\ln x)^2} \) is positive, continuous, and decreasing for \( x > 1 \).

Set up the integral \( \int_2^{\infty} \frac{1}{x (\ln x)^2} \, dx \). Use the substitution \( t = \ln x \), which implies \( dt = \frac{1}{x} dx \), so the integral becomes \( \int_{\ln 2}^{\infty} \frac{1}{t^2} \, dt \).

Evaluate the integral \( \int_{\ln 2}^{\infty} t^{-2} \, dt \) to determine if it converges. If the integral converges, then by the Integral Test, the series converges absolutely; if it diverges, the series diverges.

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Absolute and Conditional Convergence

A series converges absolutely if the series of absolute values converges. If the original series converges but not absolutely, it is conditionally convergent. Understanding this distinction helps classify the behavior of series involving alternating or positive terms.

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Percorso guidato

07:51

Choosing a Convergence Test

Integral Test for Convergence

The integral test compares a series to an improper integral of a related function. If the integral of f(x) from some point to infinity converges, then the series ∑ f(n) converges. This test is useful for series with terms like 1/[n(ln n)^p].

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Percorso guidato

07:51

Choosing a Convergence Test

Behavior of Logarithmic Functions in Series

Logarithmic terms in denominators affect convergence rates. For example, series with terms 1/[n(ln n)^p] converge or diverge depending on the exponent p. Recognizing how ln n influences the series is key to applying convergence tests correctly.

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5:26

Graphs of Logarithmic Functions

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Theory and Examples

Suppose that a₁, a₂, a₃, …, aₙ are positive numbers satisfying the following conditions:

i) a₁ ≥ a₂ ≥ a₃ ≥ …;

ii) the series a₂ + a₄ + a₈ + a₁₆ + … diverges.

Show that the series

a₁/1 + a₂/2 + a₃/3 + …

diverges.

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Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) (x + 4)ⁿ/(n3ⁿ)

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Convergent Series

Find the sums of the series in Exercises 19–24.

∑ (from n = 2 to ∞) -2/[n(n+1)]

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Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ (√(n² + n) − n)]

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