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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 10.5.3
Chapter 10, Problem 10.5.3

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

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{1}{k^2 + 1} \). We want to determine if it converges using the Comparison Test.
Recall that for large values of \( k \), the term \( \frac{1}{k^2 + 1} \) behaves similarly to \( \frac{1}{k^2} \) because the \( +1 \) becomes insignificant compared to \( k^2 \).
Choose a comparison series that is simpler but related, such as \( \sum_{k=1}^{\infty} \frac{1}{k^2} \), which is a p-series with \( p = 2 > 1 \) and is known to converge.
Use the Comparison Test by noting that \( \frac{1}{k^2 + 1} < \frac{1}{k^2} \) for all \( k \geq 1 \), so if the larger series \( \sum \frac{1}{k^2} \) converges, then the original series also converges.
Conclude that the appropriate comparison series to use is \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) because it helps establish convergence of the original series by the Comparison Test.

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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 or divergence of a series by comparing it to another 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

Choice of Comparison Series

To apply the Comparison Test effectively, select a simpler series whose convergence is known and whose terms closely resemble the original series. For ∑ 1/(k² + 1), the series ∑ 1/k² is a natural choice because for large k, 1/(k² + 1) behaves like 1/k².
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Geometric Series

Convergence of p-Series

A p-series ∑ 1/k^p converges if and only if p > 1. Since ∑ 1/k² is a p-series with p = 2, it converges. This fact helps conclude that ∑ 1/(k² + 1) also converges by comparison.
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P-Series and Harmonic 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

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

Textbook Question

51–56. {Use of Tech} Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.


aₙ₊₁ = 4aₙ + 1 a₀ = 1

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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

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


∑ (k = 1 to ∞) 1 / (k³ᐟ² + 1)

Textbook Question

The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.