Textbook Question
Using the Root Test
In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.
∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]
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 10.4.40
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)
Verified step by step guidance1
First, examine the general term of the series: \(a_n = \frac{2^n + 3^n}{3^n + 4^n}\). To determine convergence or divergence, analyze the behavior of \(a_n\) as \(n\) approaches infinity.
Identify the dominant terms in the numerator and denominator for large \(n\). Since exponential functions grow at different rates, compare \$3^n\( and \)4^n\( in the denominator, and \)2^n\( and \)3^n$ in the numerator.
Simplify the term by dividing numerator and denominator by the largest exponential term in the denominator, which is \$4^n$. This gives: \(a_n = \frac{\frac{2^n}{4^n} + \frac{3^n}{4^n}}{\frac{3^n}{4^n} + 1}\).
Rewrite the fractions inside the numerator and denominator using properties of exponents: \(\left(\frac{2}{4}\right)^n\), \(\left(\frac{3}{4}\right)^n\), and \(\left(\frac{3}{4}\right)^n\). Analyze the limit of \(a_n\) as \(n \to \infty\) by considering these terms.
If \(\lim_{n \to \infty} a_n \neq 0\), then by the Test for Divergence, the series diverges. If the limit is zero, consider applying other convergence tests such as the Comparison Test or Ratio Test to determine if the series converges.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Infinite Series and Convergence
An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. If the sum does not approach a finite value, the series diverges.
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Convergence of an Infinite Series
Limit Comparison Test
The Limit Comparison Test compares the terms of a given series with those of a known benchmark series. By taking the limit of the ratio of their terms, if the limit is a positive finite number, both series either converge or diverge together. This test is useful when terms are complex expressions.
Recommended video:
07:45Limit Comparison Test
Behavior of Exponential Terms in Series
When series terms involve exponential expressions like aⁿ, the dominant base (largest base in numerator and denominator) determines the term's behavior as n grows. Comparing dominant exponential terms helps simplify the series and decide convergence by analyzing the ratio of these dominant terms.
Recommended video:
06:00Geometric Series
Related Practice
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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Textbook Question
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)]
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
Find the first four nonzero terms of the Taylor series for the functions in Exercises 1–10.
6. (1 - x/3)^4