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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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1
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
Related Practice
Textbook Question

43–44. Periodic doses

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is


Aₙ = m + mf + ⋯ + mfⁿ⁻¹.


For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.


43.f = 0.25,m = 200 mg

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞) 2⁹k / kᵏ

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 ∞) ((-7)ᵏ / k²)

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) / (k!)

Textbook Question

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


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

Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 2 to ∞) 1 / (klnk)