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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 10.4.60
Chapter 10, Problem 10.4.60

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)

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{3^k}{k^2 + 1} \). This is an infinite series with terms involving an exponential numerator and a polynomial denominator.
Consider the behavior of the terms \( a_k = \frac{3^k}{k^2 + 1} \) as \( k \to \infty \). Since \( 3^k \) grows exponentially and \( k^2 + 1 \) grows polynomially, the numerator grows much faster than the denominator.
Recall the Divergence Test (also called the nth-term test for divergence): if \( \lim_{k \to \infty} a_k \neq 0 \), then the series diverges. Calculate \( \lim_{k \to \infty} \frac{3^k}{k^2 + 1} \).
Since \( 3^k \) grows without bound and \( k^2 + 1 \) grows much slower, the limit of \( a_k \) as \( k \to \infty \) is infinite, which is not zero.
Conclude that by the Divergence Test, the series \( \sum_{k=1}^{\infty} \frac{3^k}{k^2 + 1} \) diverges.

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Key Concepts

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

Geometric Series and Exponential Growth

A geometric series has terms with a constant ratio between consecutive terms, often involving exponential expressions like 3^k. Understanding how exponential growth compares to polynomial growth (like k²) is crucial for analyzing the behavior of series terms as k approaches infinity.
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Geometric Series

Divergence Test (Test for Divergence)

This test states that if the limit of the terms of a series does not approach zero, the series diverges. It is a quick initial check to determine if further tests are necessary, especially useful when terms involve exponential functions.
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Divergence Test (nth Term Test)

Comparison Test and Limit Comparison Test

These tests compare the given series to a known benchmark series to determine convergence or divergence. By comparing 3^k/(k²+1) to a geometric series like 3^k, one can conclude about the behavior of the original series based on the known properties of the comparison series.
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Limit Comparison Test
Related Practice
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

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! + ⋯

Textbook Question

13–52. Limits of sequences

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


{1 + cos(1⁄n)}

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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 = 3 to ∞) 1 / (k − 2)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 ∞) 1 / (k² + 4)

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?