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Chapter 9: Problem 36

Find the radius of convergence and the interval of convergence. $$ \sum_{k=0}^{\infty} \frac{(-2)^{k} x^{k+1}}{k+1} $$

Short Answer

Expert verified
Radius of convergence: \( \frac{1}{2} \), Interval: \( \left(-\frac{1}{2}, \frac{1}{2} \right) \)

Step by step solution

01

Identify the General Term

The general term of the series \( a_k \) is \( \frac{(-2)^k x^{k+1}}{k+1} \). To analyze convergence, we'll focus on the term \( \frac{(-2)^k x^{k+1}}{k+1} \).
02

Rewrite the Series

Notice that each term can be rewritten as \( x \frac{(-2x)^k}{k+1} \). This shows that the series depends on the behavior of \( (-2x)^k \).
03

Apply the Ratio Test

Use the ratio test to find the radius of convergence, \( R \).\[\lim_{k \to \infty} \left| \frac{(-2x)^{k+1}/(k+2)}{(-2x)^k/(k+1)} \right| = \lim_{k \to \infty} \left| \frac{(-2x)(k+1)}{k+2} \right| = |2x| \lim_{k \to \infty} \frac{k+1}{k+2}\]
04

Evaluate the Limit

Calculate the limit from the ratio test:\[|2x| \lim_{k \to \infty} \frac{k+1}{k+2} = |2x| \times 1 = |2x|\]The ratio test tells us the series converges if \( |2x| < 1 \).
05

Solve for x to Find Radius of Convergence

The condition \( |2x| < 1 \) implies that \( |x| < \frac{1}{2} \). Thus, the radius of convergence is \( R = \frac{1}{2} \).
06

Determine the Interval of Convergence

Considering \( |x| < \frac{1}{2} \), the series converges within \( -\frac{1}{2} < x < \frac{1}{2} \). Check the endpoints separately.- For \( x = -\frac{1}{2} \): The series becomes \( \sum \frac{-1}{k+1}^k \) which diverges.- For \( x = \frac{1}{2} \): The series similarly becomes \( \sum \frac{-1}{k+1}^k \) which also diverges.Thus, the interval of convergence is \( \left(-\frac{1}{2}, \frac{1}{2} \right) \).

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Key Concepts

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

Interval of Convergence
The interval of convergence is an essential part of understanding when a power series converges to a function. It tells us the range of values of x for which the series converges absolutely. For the given series
  • \( \sum_{k=0}^{\infty} \frac{(-2)^{k} x^{k+1}}{k+1} \), it converges absolutely within a certain interval.
  • We first determine this interval by finding the radius of convergence using the ratio test, which gives us a preliminary idea.
  • Once the radius is known, say R, we say the series converges for \( |x| < R \).
This means that the series converges for all x values between \(-R\) and \(R\). However, at the endpoints \(-R\) and \(R\), convergence is not guaranteed.
  • In fact, we must check these points individually to confirm if the series still holds.
In this specific exercise, the interval of convergence turned out to be \((-\frac{1}{2}, \frac{1}{2})\), implying it converges for all x within this range but not at the endpoints.
Ratio Test
The ratio test is a widely used tool for determining the behavior of series, especially power series. It helps us find if the series is converging absolutely or diverging.
  • For a series with general term \( a_k \), the ratio test examines the limit of the absolute values of successive terms:
\[\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\]
  • If this limit is less than 1, the series converges absolutely.
  • If it is greater than 1, the series diverges.
  • And if it equals 1, the ratio test is inconclusive.
In the given problem, applying the ratio test to the series \( \frac{(-2)^k x^{k+1}}{k+1} \) directs us to solve \( |2x| < 1 \).
  • The outcome provides the radius of convergence, which assists in determining the interval of convergence.
Conclusively, the ratio test is an indispensable method that simplifies finding convergence in power series.
Power Series
A power series is a series of the form \( \sum_{k=0}^{\infty} a_k x^k \), where \( a_k \) are constants and x is a variable. These series are powerful tools in mathematical analysis, not just because they can represent functions, but because they reveal important properties about these functions.
  • The series in our problem, \( \sum_{k=0}^{\infty} \frac{(-2)^k x^{k+1}}{k+1} \), slightly adjusts this definition because it involves \( x^{k+1} \) instead of simply \( x^k \), but is still treated as a power series.
  • The series is characterized by how its terms involve powers of x, and any power series has a specific radius and interval of convergence.
  • The key with power series is understanding how the coefficients \( a_k \) interact with x to determine whether the sum is finite, i.e., whether it converges.
  • Understanding convergence helps us approximate functions and analyze their behaviors, especially within the radius of convergence.
In short, power series provide a bridge between algebraic and analytical worlds, allowing us to explore the nature of functions.
Convergence Criteria
Convergence criteria are rules or tests used to determine whether a series converges or diverges. For power series, we often apply specific tests, like the ratio test, which serves as a primary criterion for establishing convergence.
  • The main principle here is to find a mathematical condition where the series sum approaches a finite limit.
For the given series in this exercise:
  • The convergence criteria relied on the condition \( |2x| < 1 \), stemming from the ratio test, indicating absolute convergence within that bound.
  • In broader terms, different types of convergence (absolute, conditional) can be targeted depending on the criterion being applied.
  • Absolute convergence implies that the series converges regardless of rearrangement, a stronger form of convergence than conditional convergence.
Each series and its related tests necessitate careful manipulation of terms to properly apply the criteria, ensuring accurate results, as seen in verifying both the radius and interval of convergence.

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