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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

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

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

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)

f(x) = 1 / √(1 − x²)

Textbook Question

Intervals of Convergence

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

∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]

Textbook Question

The series

eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + x⁵/5! + ⋯

converges to eˣ for all x.

a. Find a series for (d/dx)eˣ. Do you get the series for eˣ? Explain your answer.

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 ]