Textbook Question
Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.
58
views
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 10.7.17
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 ∞) ((-7)ᵏ / k²)
Verified step by step guidance1
Identify the series given: \( \sum_{k=1}^{\infty} \frac{(-7)^k}{k^2} \). We want to determine if it converges absolutely or diverges.
Recall that absolute convergence means the series \( \sum_{k=1}^{\infty} \left| a_k \right| \) converges, where \( a_k = \frac{(-7)^k}{k^2} \). So consider the series \( \sum_{k=1}^{\infty} \frac{7^k}{k^2} \).
Apply the Ratio Test, which uses the limit \( L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \). For our terms, calculate \( \left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{(-7)^{k+1} / (k+1)^2}{(-7)^k / k^2} \right| = \frac{7^{k+1}}{7^k} \cdot \frac{k^2}{(k+1)^2} = 7 \cdot \left( \frac{k}{k+1} \right)^2 \).
Evaluate the limit as \( k \to \infty \): \( L = 7 \cdot \lim_{k \to \infty} \left( \frac{k}{k+1} \right)^2 = 7 \cdot 1 = 7 \).
Interpret the result of the Ratio Test: since \( L = 7 > 1 \), the series \( \sum_{k=1}^{\infty} \frac{7^k}{k^2} \) diverges, so the original series does not converge absolutely. Because the terms do not tend to zero in absolute value, the original series diverges.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Ratio Test
The Ratio Test determines the convergence of a series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
Recommended video:
05:35
Ratio Test
Root Test
The Root Test analyzes the nth root of the absolute value of the terms in a series. If the limit of this nth root as n approaches infinity is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive.
Recommended video:
07:15Root Test
Absolute Convergence
A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence regardless of the sign of terms, which is important when applying tests like the Ratio or Root Test to series with alternating or negative terms.
Recommended video:
07:51Choosing a Convergence Test
Related Practice
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(cos(1 / k) – cos(1 / (k + 1)))
38
views
Textbook Question
43–44. Periodic doses
Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is
Aₙ = m + mf + ⋯ + mfⁿ⁻¹.
For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.
43.f = 0.25,m = 200 mg
47
views
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ
78
views
Textbook Question
What test is advisable if a series involves a factorial term?
81
views
Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(1 / √(k + 2) – 1 / √k)
55
views