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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.7.18
Chapter 10, Problem 10.7.18

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 ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]

Verified step by step guidance
1
Identify the general term of the series: \(a_n = \frac{n x^n}{4^n (n^2 + 1)}\).
Apply the Ratio Test to find the radius of convergence. Compute the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\):
\[L = \lim_{n \to \infty} \left| \frac{(n+1) x^{n+1}}{4^{n+1} ((n+1)^2 + 1)} \cdot \frac{4^n (n^2 + 1)}{n x^n} \right| = \lim_{n \to \infty} \left| \frac{n+1}{n} \cdot \frac{n^2 + 1}{(n+1)^2 + 1} \cdot \frac{|x|}{4} \right|.\]
Simplify the limit by analyzing the behavior of the rational expressions as \(n \to \infty\), which will approach 1, so \(L = \frac{|x|}{4}\).
Set the condition for convergence from the Ratio Test: \(L < 1\), which gives \(\frac{|x|}{4} < 1\), or \(|x| < 4\). This means the radius of convergence is \(R = 4\) and the interval of convergence is initially \((-4, 4)\).
Check the endpoints \(x = -4\) and \(x = 4\) by substituting into the original series and testing for convergence (using appropriate tests such as the Alternating Series Test or p-series test) to determine if the series converges absolutely, conditionally, or diverges at these points.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radius and Interval of Convergence

The radius of convergence is the distance from the center of a power series within which the series converges. The interval of convergence includes all x-values for which the series converges, possibly including endpoints. Finding these involves applying tests like the Ratio or Root Test to determine where the series converges absolutely.
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Radius of Convergence

Absolute Convergence

A series converges absolutely if the series of absolute values converges. This means ∑|a_n| converges, ensuring the original series converges regardless of term signs. Absolute convergence implies stronger convergence and often simplifies analysis of power series.
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Choosing a Convergence Test

Conditional Convergence

Conditional convergence occurs when a series converges, but not absolutely; that is, the series ∑a_n converges while ∑|a_n| diverges. This typically happens at the endpoints of the interval of convergence and requires careful testing, such as the Alternating Series Test, to confirm.
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Choosing a Convergence Test
Related Practice
Textbook Question

In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.

∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]

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Textbook Question

In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.

aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ

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Textbook Question

Direct Comparison Test

In Exercises 1–8, use the Direct Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) (√n + 1) / (√(n² + 3))

1
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Textbook Question

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 2 to ∞) [(ln n / n)³]

Textbook Question

Applying the Integral Test

Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑ (from n = 2 to ∞) ln(n²) / n

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Textbook Question

Find the value of b for which

1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.