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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 10.5.39
Chapter 10, Problem 10.5.39

38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.
39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 1)

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First, recognize the general term of the series: \(a_k = \frac{1}{k^2 + 2k + 1}\). Notice that the denominator can be factored as \((k + 1)^2\).
Rewrite the series as \(\sum_{k=1}^\infty \frac{1}{(k+1)^2}\). This is similar to the p-series \(\sum \frac{1}{k^p}\) with \(p=2\).
Use the Comparison Test by comparing \(a_k\) to \(b_k = \frac{1}{k^2}\). Since \(\frac{1}{(k+1)^2} < \frac{1}{k^2}\) for all \(k\), and we know \(\sum \frac{1}{k^2}\) converges (p-series with \(p=2 > 1\)), the original series converges by comparison.
Alternatively, apply the Limit Comparison Test with \(b_k = \frac{1}{k^2}\). Compute the limit \(L = \lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} \frac{1/(k+1)^2}{1/k^2} = \lim_{k \to \infty} \left( \frac{k}{k+1} \right)^2\).
Since \(L = 1\) (a finite positive number), both series either converge or diverge together. Because \(\sum \frac{1}{k^2}\) converges, the original series converges as well.

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Key Concepts

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

Comparison Test

The Comparison Test determines the convergence of a series by comparing it to a second series with known behavior. If the terms of the given series are smaller than those of a convergent series, it also converges. Conversely, if the terms are larger than those of a divergent series, it diverges.
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Direct Comparison Test

Limit Comparison Test

The Limit Comparison Test compares two series by taking the limit of the ratio of their terms. If the limit is a positive finite number, both series either converge or diverge together. This test is useful when direct comparison is difficult but the series have similar term behavior.
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Limit Comparison Test

Recognizing and Simplifying Series Terms

Simplifying the general term of a series can reveal its form and help identify a comparable series. For example, rewriting 1/(k² + 2k + 1) as 1/(k+1)² shows it resembles a p-series with p=2, which is known to converge. Recognizing such forms aids in applying convergence tests effectively.
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Geometric Series
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