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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 10.7.35
Chapter 10, Problem 10.7.35

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (2k + 1)! / (k!)²

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1
First, identify the general term of the series: \(a_k = \frac{(2k + 1)!}{(k!)^2}\).
To determine convergence, consider using the Ratio Test, which is effective for factorial expressions. The Ratio Test states to compute \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\).
Write out the ratio \(\frac{a_{k+1}}{a_k} = \frac{(2(k+1) + 1)!}{((k+1)!)^2} \times \frac{(k!)^2}{(2k + 1)!}\) and simplify the factorial expressions carefully.
Evaluate the limit \(L\) as \(k\) approaches infinity. If \(L < 1\), the series converges absolutely; if \(L > 1\), the series diverges; if \(L = 1\), the test is inconclusive.
Based on the result of the Ratio Test, conclude whether the series converges absolutely, converges conditionally, or diverges.

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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

A series converges absolutely if the series of absolute values converges. If the original series converges but not absolutely, it converges conditionally. Understanding these distinctions 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 ratio of consecutive terms. If the limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
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Ratio Test

Factorials and Growth Rates

Factorials grow very rapidly, often dominating polynomial terms. Comparing factorial expressions using simplification or the Ratio Test helps analyze the behavior of series terms and determine convergence or divergence.
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Factorials
Related Practice
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 ∞) (k² / 4ᵏ)

Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k^(–1/5)

Textbook Question

9–15. Geometric sums Evaluate each geometric sum.


{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ

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Textbook Question

Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?

Textbook Question

39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.


∑ (k = 1 to ∞) (−1)ᵏ / k⁵

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Textbook Question

17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.

∑ (k = 1 to ∞) 1 / (∛(5k + 3))