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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.4.5
Chapter 10, Problem 10.4.5

Direct Comparison Test
In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) cos²n / n^(3/2)

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1
Identify the given series: \( \sum_{n=1}^{\infty} \frac{\cos^{2} n}{n^{3/2}} \). Notice that the terms involve \( \cos^{2} n \) divided by \( n^{3/2} \).
Recall the Direct Comparison Test: To use it, we need to compare the given series to a known benchmark series \( \sum b_n \) where \( b_n \) is positive and whose convergence behavior is known.
Since \( \cos^{2} n \) oscillates between 0 and 1, we know \( 0 \leq \cos^{2} n \leq 1 \) for all \( n \). Therefore, each term of the series satisfies \( 0 \leq \frac{\cos^{2} n}{n^{3/2}} \leq \frac{1}{n^{3/2}} \).
Compare the given series to the p-series \( \sum_{n=1}^{\infty} \frac{1}{n^{3/2}} \). Since \( p = \frac{3}{2} > 1 \), this p-series converges.
By the Direct Comparison Test, since \( 0 \leq \frac{\cos^{2} n}{n^{3/2}} \leq \frac{1}{n^{3/2}} \) and \( \sum \frac{1}{n^{3/2}} \) converges, the original series \( \sum \frac{\cos^{2} n}{n^{3/2}} \) also converges.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Direct Comparison Test

The Direct Comparison Test determines the convergence or divergence of a series by comparing it to another series with known behavior. If 0 ≤ a_n ≤ b_n for all n and ∑b_n converges, then ∑a_n also converges. Conversely, if a_n ≥ b_n ≥ 0 and ∑b_n diverges, then ∑a_n diverges.
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Direct Comparison Test

Behavior of the Cosine Function

The term cos²(n) oscillates between 0 and 1 for all integer n. Since cos²(n) is always non-negative and bounded above by 1, it can be used to compare the given series to a simpler series by replacing cos²(n) with its upper bound.
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Graph of Sine and Cosine Function

p-Series Test

A p-series ∑ 1/n^p converges if and only if p > 1 and diverges otherwise. In this problem, the denominator n^(3/2) corresponds to p = 3/2, which is greater than 1, so the p-series converges. This fact helps in comparing the given series to a convergent p-series.
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P-Series and Harmonic Series
Related Practice
Textbook Question

If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.

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Textbook Question

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 ∞) (tanh n) / n²

Textbook Question

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 ∞) [(x + 1)²ⁿ] / 9ⁿ

Textbook Question

Convergence and Divergence

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

aₙ = 8^(1/n)

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Textbook Question

Convergence and Divergence

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

aₙ = (2n + 2)! / (2n − 1)!

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Textbook Question

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⁻¹ / (n² + 2n + 1)]