Textbook Question
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ / (1 + √n)]
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Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.2.13
Series with Geometric Terms
In Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.
∑ (from n = 0 to ∞) [(1 / 2ⁿ) + ((-1)ⁿ / 5ⁿ)]
Verified step by step guidance1
Identify the general term of the series: \( a_n = \frac{1}{2^n} + \frac{(-1)^n}{5^n} \). This series is the sum of two separate geometric series.
Write out the first eight terms by substituting \( n = 0, 1, 2, \ldots, 7 \) into the general term: \( a_0, a_1, a_2, \ldots, a_7 \). This helps visualize how the series starts.
Recognize that each part is a geometric series: the first with common ratio \( r_1 = \frac{1}{2} \), and the second with common ratio \( r_2 = -\frac{1}{5} \). Both have initial terms \( a_0^{(1)} = 1 \) and \( a_0^{(2)} = 1 \) respectively.
Check the absolute values of the common ratios to determine convergence: since \( |r_1| < 1 \) and \( |r_2| < 1 \), both geometric series converge.
Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to find the sum of each series separately, then add these sums to get the total sum of the original series.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Series
A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^n, where a is the first term and r is the common ratio. Understanding this helps in identifying and summing series with terms like (1/2)^n or (-1/5)^n.
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Geometric Series
Sum of an Infinite Geometric Series
An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. Its sum is given by S = a / (1 - r). This formula is essential for finding the sum of each geometric component in the series when it converges.
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06:00Geometric Series
Series Divergence and Convergence
A series converges if its partial sums approach a finite limit; otherwise, it diverges. For series composed of multiple parts, each part must be analyzed separately. Recognizing divergence or convergence is key to determining whether the sum exists or not.
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06:52Convergence of an Infinite Series
Related Practice
Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)
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Textbook Question
Finding Terms of a Sequence
Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.
aₙ = 2 + (-1)ⁿ
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Textbook Question
Determining Convergence or Divergence
Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
∑ (from n=1 to ∞) (1 + 1/n)ⁿ
Textbook Question
Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]
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Textbook Question
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4