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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 10.4.47e
Chapter 10, Problem 10.4.47e

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. If ∑ k⁻ᵖ converges, then ∑ k⁻ᵖ⁺⁰.⁰⁰¹ converges.

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Recall the p-series test: The series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges if and only if \( p > 1 \).
Given that \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges, it implies that \( p > 1 \).
Now consider the series \( \sum_{k=1}^{\infty} \frac{1}{k^{p + 0.001}} \). Since \( p + 0.001 > p > 1 \), this new exponent is also greater than 1.
By the p-series test again, since the exponent \( p + 0.001 \) is greater than 1, the series \( \sum_{k=1}^{\infty} \frac{1}{k^{p + 0.001}} \) also converges.
Therefore, the statement is true because increasing the exponent in a convergent p-series to a slightly larger value still results in a convergent series.

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Key Concepts

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

p-Series Test for Convergence

The p-series test states that the series ∑ 1/k^p converges if and only if p > 1. This test helps determine whether a series with terms involving powers of k converges or diverges based on the value of p.
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P-Series and Harmonic Series

Comparison Test for Series

The comparison test allows us to determine convergence by comparing a given series to another series with known behavior. If a series with larger terms converges, then a series with smaller terms also converges, and vice versa.
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Direct Comparison Test

Effect of Small Changes in the Exponent on Convergence

Slightly increasing the exponent p in a p-series (e.g., from p to p + 0.001) affects convergence because the series terms decrease faster. If the original series converges, increasing p will also result in convergence, but if it diverges, the new series may still diverge.
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Related Practice
Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


e. ∑ (k = 1 to ∞) (π / e)⁻ᵏ is a convergent geometric series.

Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

f.If the sequence {aₙ} diverges, then the sequence {0.000001aₙ} diverges.

Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


f. If the series ∑ (k = 1 to ∞) aᵏ converges and |a| < |b|, then the series ∑ (k = 1 to ∞) bᵏ converges.

Textbook Question

87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


d. If ∑ pᵏ diverges, then ∑ (p + 0.001)ᵏ diverges, for a fixed real number p.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

f. If lim (k → ∞) aₖ = 0, then ∑ aₖ converges."

Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


d.If {aₙ} = {1, ½, ⅓, ¼, ⅕, …} and

{bₙ} = {1, 0, ½, 0, ⅓, 0, ¼, 0, …},

then limₙ→∞aₙ = limₙ→∞bₙ.

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