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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 10.6.1
Capitolo 10, Problema 10.6.1

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 = 1 to ∞) [(-1)ⁿ⁺¹ (1 / n^(3/2))]

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Identify the general term of the series: \(a_n = \frac{1}{n^{3/2}}\) and the alternating factor is \((-1)^{n+1}\), which means the series is alternating in sign.
Check the conditions of the Alternating Series Test (Leibniz Test): (1) The terms \(a_n\) must be positive, (2) \(a_n\) must be decreasing, and (3) \(\lim_{n \to \infty} a_n = 0\).
Verify that \(a_n = \frac{1}{n^{3/2}}\) is positive for all \(n\) and that it decreases as \(n\) increases because the denominator grows larger.
Calculate the limit \(\lim_{n \to \infty} \frac{1}{n^{3/2}}\) to confirm it approaches zero, satisfying the third condition of the Alternating Series Test.
Since all conditions are met, conclude that the alternating series converges by the Alternating Series Test.

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Alternating Series Test

The Alternating Series Test determines convergence of series whose terms alternate in sign. It requires that the absolute value of terms decreases monotonically to zero. If these conditions hold, the series converges, even if it does not converge absolutely.
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Alternating Series Test

Absolute vs Conditional Convergence

A series converges absolutely if the series of absolute values converges. If the original alternating series converges but not absolutely, it is conditionally convergent. Understanding this distinction helps classify the nature of convergence.
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Choosing a Convergence Test

p-Series and Convergence

A p-series has the form ∑ 1/n^p and converges if p > 1. In this problem, the terms involve 1/n^(3/2), where p = 3/2 > 1, indicating the absolute value series converges, which implies absolute convergence of the alternating series.
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P-Series and Harmonic Series
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Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=1 to ∞) [(-1)ⁿ n² e⁻ⁿ]

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Finding Taylor Polynomials

In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.

f(x) = √(1 − x),a = 0

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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 = 2 + (0.1)ⁿ

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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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In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001.

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

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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 ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]

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