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 10.5.2

Capitolo 10, Problema 10.5.2

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

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Identify the general term of the series: \(a_n = \frac{(-1)^n (n + 2)}{3^n}\).

Set up the Ratio Test by considering the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).

Substitute \(a_{n+1}\) and \(a_n\) into the limit expression: \(L = \lim_{n \to \infty} \left| \frac{(-1)^{n+1} (n + 3) / 3^{n+1}}{(-1)^n (n + 2) / 3^n} \right|\).

Simplify the expression inside the limit by canceling common factors and absolute values, noting that \(|(-1)^n| = 1\), to get \(L = \lim_{n \to \infty} \frac{n + 3}{n + 2} \cdot \frac{1}{3}\).

Evaluate the limit \(L\) and use the Ratio Test criteria: if \(L < 1\), the series converges absolutely; if \(L > 1\), it 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.

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05:35

Ratio Test

Absolute Convergence

A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence implies convergence regardless of the sign of the terms, which is stronger than conditional convergence where the series converges but not absolutely.

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07:51

Choosing a Convergence Test

Infinite Series with Alternating Terms

An infinite series with alternating terms includes factors like (-1)^n that cause the terms to alternate in sign. Such series may converge conditionally or absolutely, and tests like the Ratio Test help determine the nature of their convergence.

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06:52

Convergence of an Infinite Series

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Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 0 to ∞) (2x)ⁿ

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Are there any values of x for which ∑ (from n=1 to ∞) (1 / nˣ) converges? Give reasons for your answer.

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Determining Convergence or Divergence

In Exercises 1–14, determine whether the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test.

∑ (from n = 2 to ∞) [(-1)ⁿ⁺¹ (1 / ln n)]

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Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²]

(Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)

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Recursively Defined Sequences

In Exercises 101–108, assume that each sequence converges and find its limit.

a₁ = 2,aₙ₊₁ = 72 / (1 + aₙ)