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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.7.47
Chapter 10, Problem 10.7.47

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
1 / 1! + 4 / 2! + 9 / 3! + 16 / 4! + ⋯

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1
Identify the general term of the series. Notice the pattern in the numerator and denominator: the numerator is \(n^2\) and the denominator is \(n!\). So the general term can be written as \(a_n = \frac{n^2}{n!}\).
To determine convergence, consider the absolute value of the terms. Since all terms are positive, absolute convergence is the same as convergence here.
Apply the Ratio Test, which is useful for series involving factorials. The Ratio Test states to compute the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
Calculate the ratio \(\frac{a_{n+1}}{a_n} = \frac{\frac{(n+1)^2}{(n+1)!}}{\frac{n^2}{n!}} = \frac{(n+1)^2}{(n+1)!} \times \frac{n!}{n^2} = \frac{(n+1)^2}{(n+1) \times n^2} = \frac{n+1}{n^2}\).
Evaluate the limit \(L = \lim_{n \to \infty} \frac{n+1}{n^2}\). Since this limit approaches 0, which is less than 1, the Ratio Test tells us the series converges absolutely.

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Key Concepts

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

Absolute and Conditional Convergence

Absolute convergence occurs when the series of absolute values converges, ensuring the original series converges regardless of term signs. Conditional convergence happens when the series converges but not absolutely, meaning the series converges only due to the arrangement of terms. Understanding these helps classify the behavior of infinite series.
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Choosing a Convergence Test

Ratio Test

The Ratio Test determines convergence by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges; if equal to one, the test is inconclusive. This test is especially useful for series involving factorials or exponentials.
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Ratio Test

Factorials and Growth Rates

Factorials grow faster than polynomial expressions, which affects the convergence of series with terms involving factorials in the denominator. Recognizing how factorial growth compares to polynomial or exponential growth helps in applying convergence tests effectively, as factorial terms often lead to rapid decay of series terms.
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Factorials
Related Practice
Textbook Question

21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

aₙ₊₁ = 2aₙ; a₁ = 2

Textbook Question

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) 1 / ( (3k + 1)(3k + 4) )

Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) 20 / (∛k + √k)

Textbook Question

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) 3ᵏ / (k² + 1)

Textbook Question

13–52. Limits of sequences

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


{1 + cos(1⁄n)}

1
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Textbook Question

For what values of p does the series ∑ (k = 10 to ∞) 1 / kᵖ converge (initial index is 10)? For what values of p does it diverge?