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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.PE.51
Chapter 10, Problem 10.PE.51

Power Series
In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.
∑ (from n = 1 to ∞) xⁿ/nⁿ

Verified step by step guidance
1
Identify the given power series: \(\displaystyle \sum_{n=1}^{\infty} \frac{x^{n}}{n^{n}}\).
To find the radius of convergence, apply the Root Test by considering the nth root of the absolute value of the general term: \(\displaystyle \lim_{n \to \infty} \sqrt[n]{\left| \frac{x^{n}}{n^{n}} \right|} = \lim_{n \to \infty} \frac{|x|}{n}\).
Evaluate the limit: since \(\lim_{n \to \infty} \frac{|x|}{n} = 0\) for all real \(x\), the Root Test implies the series converges for all \(x\) (radius of convergence \(R = \infty\)).
For absolute convergence, consider the series with absolute values: \(\sum_{n=1}^{\infty} \frac{|x|^{n}}{n^{n}}\). Since the radius of convergence is infinite, the series converges absolutely for all real \(x\).
Since the series converges absolutely everywhere, there are no values of \(x\) for which the series converges only conditionally.

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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 defines the distance from the center of a power series within which the series converges. The interval of convergence is the set of all x-values for which the series converges, including endpoints if applicable. These are typically found using the Ratio or Root Test.
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Radius of Convergence

Absolute Convergence

A series converges absolutely if the series of absolute values converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it often simplifies analysis since absolute convergence implies regular convergence.
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Choosing a Convergence Test

Conditional Convergence

Conditional convergence occurs when a series converges but does not converge absolutely. This means the series converges only because of the specific arrangement or signs of its terms, and rearranging terms can affect convergence.
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Choosing a Convergence Test
Related Practice
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

Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) (csch n)xⁿ

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 = 1 + (0.9)ⁿ

1
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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!

1
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Textbook Question

Maclaurin Series

Find Taylor series at x = 0 for the functions in Exercises 63–70.

cos (x³/√5)