Textbook Question
Define the remainder of an infinite series.
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.5.53
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0
Verified step by step guidance1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{1}{k^{1+p}} \) where \( p > 0 \). This is a p-series, a common type of series in calculus.
Recall the p-series test: A series of the form \( \sum_{k=1}^{\infty} \frac{1}{k^q} \) converges if and only if \( q > 1 \).
Compare the exponent in the denominator of the given series, which is \( 1 + p \), with 1. Since \( p > 0 \), then \( 1 + p > 1 \).
Apply the p-series test: Because \( 1 + p > 1 \), the series \( \sum_{k=1}^{\infty} \frac{1}{k^{1+p}} \) converges.
Conclude that the series converges for all \( p > 0 \) based on the p-series convergence criterion.
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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 k.
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P-Series and Harmonic Series
Comparison Test
The comparison test involves comparing a given series to a known benchmark series to determine convergence or divergence. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.
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09:25Direct Comparison Test
Convergence of Infinite Series
Understanding convergence means determining whether the sum of infinitely many terms approaches a finite limit. Tests like the p-series test and comparison test help decide if the infinite sum converges or diverges, which is crucial for analyzing series behavior.
Recommended video:
06:52Convergence of an Infinite Series
Related Practice
Textbook Question
1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 10 to ∞) 1 / (k − 9)⁵
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹(10n⁄(10n + 4))}
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Textbook Question
1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.
∑ (from k = 3 to ∞) (2k²) / (k² − k − 2)
Textbook Question
6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.
{(−0.7)ⁿ}
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Textbook Question
Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k.
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