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 ∞) (k² / 4ᵏ)
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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 10.4.23
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 = 1 to ∞) k^(–1/5)
Verified step by step guidance1
Identify the series given: \( \sum_{k=1}^{\infty} k^{-\frac{1}{5}} \). This is a series with terms of the form \( k^{-p} \), where \( p = \frac{1}{5} \).
Recognize that this is a p-series, which has the general form \( \sum_{k=1}^{\infty} \frac{1}{k^p} \). The convergence of a p-series depends on the value of \( p \).
Recall the p-series test: A p-series \( \sum \frac{1}{k^p} \) converges if and only if \( p > 1 \), and diverges otherwise.
Since \( p = \frac{1}{5} = 0.2 \), which is less than 1, the p-series test indicates that the series diverges.
Optionally, you could confirm this result using the Divergence Test by checking \( \lim_{k \to \infty} k^{-\frac{1}{5}} \). If the limit is not zero, the series diverges. Here, the limit is zero, so the Divergence Test is inconclusive, but the p-series test already gives the answer.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Divergence Test
The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, the series diverges. It is a quick initial check to determine if a series cannot converge, but if the limit is zero, the test is inconclusive.
Recommended video:
05:44
Divergence Test (nth Term Test)
Integral Test
The Integral Test relates the convergence of a series to the convergence of an improper integral. If the function f(k) = a_k is positive, continuous, and decreasing for k ≥ 1, then the series ∑a_k converges if and only if the integral from 1 to infinity of f(x) dx converges.
Recommended video:
07:25Integral Test
p-series Test
A p-series is a series of the form ∑ 1/k^p. It converges if and only if p > 1 and diverges otherwise. This test is useful for quickly determining the behavior of series with terms involving powers of k.
Recommended video:
04:30P-Series and Harmonic Series
Related Practice
Textbook Question
9–15. Geometric sums Evaluate each geometric sum.
{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ
1
views
Textbook Question
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)
1
views
Textbook Question
40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) (1 + 2 / 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.
∑ (from k = 1 to ∞) (2k + 1)! / (k!)²
Textbook Question
17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))