Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.4.5

Chapter 10, Problem 10.4.5

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?

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Recognize that the given series is a p-series of the form \(\sum_{k=10}^{\infty} \frac{1}{k^p}\), where \(p\) is a real number parameter.

Recall the p-series convergence test: a p-series \(\sum_{k=1}^{\infty} \frac{1}{k^p}\) converges if and only if \(p > 1\) and diverges otherwise.

Note that changing the starting index from 1 to 10 does not affect the convergence or divergence of the series because convergence depends on the behavior of the tail of the series.

Therefore, apply the p-series test to conclude that the series converges if \(p > 1\) and diverges if \(p \leq 1\).

Summarize the result: the series \(\sum_{k=10}^{\infty} \frac{1}{k^p}\) converges for all \(p > 1\) and diverges for all \(p \leq 1\).

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Key Concepts

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

p-Series Test

The p-series test determines the convergence of series of the form ∑ 1/k^p. Such a series converges if and only if p > 1, and diverges otherwise. This test is fundamental for analyzing series with terms involving powers of the index.

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence criteria helps determine whether the sum of infinitely many terms results in a finite value or not.

Effect of the Starting Index on Convergence

Changing the starting index of a series (e.g., from 1 to 10) does not affect its convergence or divergence. Convergence depends on the behavior of terms as k approaches infinity, so initial finite terms do not influence the overall convergence.

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Choosing a Convergence Test

Related Practice

Textbook Question

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 ∞) 1 / ( (3k + 1)(3k + 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 ∞) 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

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)

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