Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problem 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

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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.

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.

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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