Textbook Question
Make up an infinite series of nonzero terms whose sum is
b. −3
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 10.7.42
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ ( n / (n + 1) )ⁿ^ ² ] xⁿ (Hint: Apply the Root Test.)
Verified step by step guidance1
Identify the general term of the series as \(a_n = \left( \frac{n}{n+1} \right)^{n^2} x^n\).
Recall that the Root Test involves computing the limit \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\) to determine convergence.
Apply the \(n\)th root to the absolute value of the general term: \(\sqrt[n]{|a_n|} = \sqrt[n]{\left( \frac{n}{n+1} \right)^{n^2} |x|^n} = \left( \frac{n}{n+1} \right)^n |x|\).
Evaluate the limit \(L = \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^n |x|\). Recognize that \(\left( \frac{n}{n+1} \right)^n\) can be rewritten and simplified using properties of limits and exponentials.
Use the Root Test conclusion: the series converges if \(L < 1\), so set up the inequality involving \(|x|\) and solve for the radius of convergence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radius of Convergence
The radius of convergence of a power series is the distance from the center of the series within which the series converges absolutely. It determines the interval on the x-axis where the series represents a valid function. Finding this radius helps understand the domain of convergence for the series.
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Radius of Convergence
Root Test
The Root Test is a method to determine the convergence of an infinite series by examining the nth root of the absolute value of its terms. Specifically, for a series ∑a_n, if the limit of the nth root of |a_n| is L, the series converges if L < 1 and diverges if L > 1. It is especially useful for series with terms raised to the nth power.
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07:15Root Test
Power Series and General Term
A power series is an infinite sum of terms in the form a_n x^n, where a_n depends on n. Understanding the general term, including coefficients and powers of n, is crucial for applying convergence tests. Here, the term involves (n/(n+1))^(n^2), which affects the behavior of the series and its convergence.
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05:58Intro to Power Series
Related Practice
Textbook Question
Repeating Decimals
Express each of the numbers in Exercises 23–30 as the ratio of two integers.
3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...
Textbook Question
Finding a Sequence’s Formula
In Exercises 13–30, find a formula for the nth term of the sequence.
1, -4, 9, -16, 25, …Squares of the positive integers, with alternating signs
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Textbook Question
Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 1 to ∞) (1 − 1/n)ⁿ
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Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 0 to ∞) (x + 5)ⁿ
Textbook Question
Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.
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