Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.7.7

Chapter 10, Problem 10.7.7

What test is advisable if a series involves a factorial term?

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When a series involves factorial terms, it is often advisable to use the Ratio Test to determine convergence or divergence.

The Ratio Test involves examining the limit of the absolute value of the ratio of consecutive terms: \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).

For a series with terms involving factorials, this test simplifies the factorial expressions because factorials cancel nicely when taking ratios of consecutive terms.

If the limit from the Ratio Test is less than 1, the series converges absolutely; if it is greater than 1, the series diverges; and if it equals 1, the test is inconclusive.

Therefore, the Ratio Test is the most effective and commonly used test for series with factorial terms.

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Key Concepts

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

Factorial Terms in Series

Factorial terms, denoted by n!, grow very rapidly and often appear in series related to permutations or combinations. Recognizing factorials in series helps determine the appropriate convergence test, as their growth rate affects the behavior of the series.

Ratio Test

The Ratio Test is used to determine the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. It is especially effective for series with factorials or exponential terms because it simplifies the factorial expressions and reveals convergence behavior.

Convergence of Infinite Series

Understanding when an infinite series converges or diverges is fundamental in calculus. Convergence means the sum approaches a finite value, and tests like the Ratio Test help decide this, particularly when terms involve complex expressions like factorials.

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Convergence of an Infinite Series

Related Practice

Textbook Question

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(−1)ⁿ / 2ⁿ}

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Textbook Question

84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.

a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b.Use analytical methods to find the limit of the sequence.

aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3

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Textbook Question

What comparison series would you use with the Limit Comparison Test to determine whether ∑ (k = 1 to ∞) (k² + k + 5) / (k³ + 3k + 1) converges?

Textbook Question

{Use of Tech} For what value of r does

∑ (k = 3 to ∞) r²ᵏ = 10?

Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ

Textbook Question

What comparison series would you use with the Comparison Test to determine whether

∑ (k = 1 to ∞) 1 / (k² + 1) converges?

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