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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.4.20
Capitolo 10, Problema 10.4.20

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 + cos n) / n²

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1
Identify the general term of the series: \(a_n = \frac{1 + \cos n}{n^2}\).
Recall that \(\cos n\) oscillates between \(-1\) and \(1\), so \(1 + \cos n\) is bounded between \(0\) and \(2\).
Since \(a_n\) behaves roughly like \(\frac{\text{bounded term}}{n^2}\), compare it to the convergent p-series \(\sum \frac{1}{n^2}\), where \(p=2 > 1\).
Apply the Comparison Test: because \(0 \leq a_n \leq \frac{2}{n^2}\) and \(\sum \frac{2}{n^2}\) converges, the original series converges by the Comparison Test.
Conclude that the series \(\sum_{n=1}^\infty \frac{1 + \cos n}{n^2}\) converges absolutely, since the absolute value \(|a_n| \leq \frac{2}{n^2}\) also forms a convergent p-series.

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Convergence and Divergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges. Understanding this distinction is fundamental to analyzing series behavior.
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Convergence of an Infinite Series

Comparison Test for Series

The Comparison Test involves comparing a given series to a second series whose convergence behavior is known. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test helps determine convergence by bounding.
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Direct Comparison Test

Behavior of Trigonometric Functions in Series

Trigonometric functions like cosine oscillate between fixed bounds, affecting the terms of a series. When combined with a decreasing denominator (like n²), the oscillations are controlled, often allowing convergence tests to focus on the dominant term's behavior.
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Introduction to Trigonometric Functions
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Domanda del libro di testo

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 ∞) [(2 + (−1)ⁿ) / 1.25ⁿ]

Domanda del libro di testo

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 0 to ∞) [(2n + 3)(2ⁿ + 3) / (3ⁿ + 2)]

Domanda del libro di testo

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 2 to ∞) [(3n)! / (n!)³]

Domanda del libro di testo

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 / (1 + 2² + 3² + ⋯ + n²)

Domanda del libro di testo

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=3 to ∞) [2n² / n²ⁿ]

Domanda del libro di testo

Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 31–38.

(tan⁻¹x)²