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Chapter 9: Problem 6

(a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely, (c) conditionally? $$\sum_{n=0}^{\infty}(2 x)^{n}$$

Short Answer

Expert verified
(a) Radius: \( \frac{1}{2} \), Interval: \( -\frac{1}{2} < x < \frac{1}{2} \). (b) Absolutely: \( -\frac{1}{2} < x < \frac{1}{2} \). (c) Conditionally: None.

Step by step solution

01

Identify the series type

The series given is \( \sum_{n=0}^{\infty} (2x)^n \), which is a geometric series. The generic form of a geometric series is \( \sum_{n=0}^{\infty} r^n \), where \( r \) is the common ratio.
02

Determine the common ratio

From the series \( \sum_{n=0}^{\infty} (2x)^n \), we identify that the common ratio \( r \) is \( 2x \).
03

Find the radius of convergence

For a geometric series \( \sum_{n=0}^{\infty} r^n \), the series converges when \( |r| < 1 \). Thus, for our series, the condition for convergence is \( |2x| < 1 \). This simplifies to \( |x| < \frac{1}{2} \). Therefore, the radius of convergence \( R \) is \( \frac{1}{2} \).
04

Determine the interval of convergence

Given \( |x| < \frac{1}{2} \), the interval of convergence is \( -\frac{1}{2} < x < \frac{1}{2} \).
05

Check for absolute convergence

A geometric series converges absolutely within the interval of convergence. Thus, for \( -\frac{1}{2} < x < \frac{1}{2} \), the series converges absolutely.
06

Check for conditional convergence

Since a geometric series either converges absolutely or diverges, there is no interval for conditional convergence. For \( x = -\frac{1}{2} \) and \( x = \frac{1}{2} \), the series diverges completely as it violates the \( |r| < 1 \) condition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Series
A geometric series is a series where each term is a constant multiple, known as the common ratio, of the previous term. This type of series has the general form \( \sum_{n=0}^{\infty} r^n \), where \( r \) represents the common ratio.
When dealing with geometric series, knowing the common ratio is crucial because it determines whether the series converges.
If \( |r| < 1 \), the geometric series converges. Otherwise, it diverges.
  • Convergence happens because as \( n \) becomes very large, \( r^n \) approaches zero, making the sum of the series converge to a finite number.
  • Divergence occurs when \( |r| \geq 1 \), as \( r^n \) either does not approach zero or oscillates indefinitely.
This simple yet powerful structure of geometric series allows us to easily determine the radius of convergence, which is often essential in understanding series behavior around specific points.
Absolute Convergence
Absolute convergence occurs when the series of absolute values of its terms also converges.
In simpler terms, if \( \sum |a_n| \) is convergent, then \( \sum a_n \) is said to converge absolutely. This is a stronger form of convergence and implies the original series converges as well.
For our geometric series \( \sum_{n=0}^{\infty} (2x)^n \), absolute convergence is observed when \( |2x| < 1 \), or equivalently \( |x| < \frac{1}{2} \).
  • This means that within the interval of convergence, from \( -\frac{1}{2} \) to \( \frac{1}{2} \), the series of terms possesses the property that its absolute values also form a convergent series.
  • Absolute convergence ensures stability in the behavior of the series and is often a preferred form of convergence because it is more mathematically "safe" to work with.
It is important to check for absolute convergence as it guarantees no unexpected behavior in the series' sum over its interval of convergence.
Conditional Convergence
Conditional convergence is a situation where the series \( \sum a_n \) converges, but the series formed by the absolute values, \( \sum |a_n| \), does not converge.
This type of convergence is usually not observed in geometric series like \( \sum_{n=0}^{\infty} (2x)^n \), since their nature makes them either absolutely convergent or divergent.
In our exercise, the series converges absolutely for \( -\frac{1}{2} < x < \frac{1}{2} \).
  • There is no conditional convergence because the series cannot converge conditionally due to the fact that geometric series either converge completely (absolutely) or not at all.
  • At the boundary points \( x = -\frac{1}{2} \) and \( x = \frac{1}{2} \), the series diverges because it fails the \( |r| < 1 \) condition, confirming no conditional convergence at these extremes.
Understanding the absence of conditional convergence in geometric series reinforces their straightforward nature and helps clarify when this type of convergence might be considered.

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