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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 10.7.65a
Chapter 10, Problem 10.7.65a

Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
a. Converges absolutely for x = 1

Verified step by step guidance
1
Identify the center of the power series, which is at \(x = 2\), since the series is given as \(\sum a_n (x - 2)^n\).
Use the information about convergence and divergence at specific points to find the radius of convergence \(R\). The series converges at \(x = -1\) and diverges at \(x = 6\).
Calculate the distances from the center to these points: \(|-1 - 2| = 3\) and \(|6 - 2| = 4\). Since the series converges at \(x = -1\) (distance 3) and diverges at \(x = 6\) (distance 4), the radius of convergence \(R\) satisfies \(3 \leq R < 4\).
Determine whether the point \(x = 1\) lies inside the radius of convergence by calculating \(|1 - 2| = 1\), which is less than \(R\). Therefore, the series converges at \(x = 1\).
Recall that convergence inside the radius of convergence is absolute convergence for power series, so the series converges absolutely at \(x = 1\).

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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 ∑ aₙ(x − c)ⁿ is the distance from the center c within which the series converges. It is determined by the interval where the series converges absolutely. Knowing convergence at specific points helps estimate this radius.
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Radius of Convergence

Interval of Convergence

The interval of convergence is the set of x-values for which the power series converges. It is centered at c and extends to the radius of convergence on both sides, but convergence at endpoints must be checked separately.
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Interval of Convergence

Absolute vs Conditional Convergence

Absolute convergence means the series of absolute values ∑ |aₙ(x − c)ⁿ| converges, guaranteeing convergence of the original series. Conditional convergence occurs when the series converges but not absolutely. This distinction affects conclusions about convergence at given points.
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Choosing a Convergence Test
Related Practice
Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence.

∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]

Textbook Question

Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.

a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]

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Textbook Question

(Continuation of Exercise 61.) Use the result in Exercise 61 to determine which of the following series converge and which diverge. Support your answer in each case.

a. ∑ (from n=2 to ∞) [1 / (n ln n)]

Textbook Question

The series

sec x = 1 + x²/2 + 5x⁴/24 + 61x⁶/720 + 277x⁸/8064 + ⋯

converges to sec x for −π/2 < x < π/2.

a. Find the first five terms of a power series for the function ln|sec x + tan x|. For what values of x should the series converge?

Textbook Question

Determining Convergence of Sequences

Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.


aₙ = (-4)ⁿ/n!

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Textbook Question

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence.

∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]