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Chapter 9: Problem 5

Find the interval of convergence of the given series. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} x^{2 n} $$

Short Answer

Expert verified
The interval of convergence is \([-1, 1]\).

Step by step solution

01

Understanding the Series

The given series is \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} x^{2n} \). This is a power series with general term \( a_n x^{2n} \) where \( a_n = \frac{(-1)^n}{n+1} \). We need to use the ratio test to find the interval of convergence for this power series.
02

Applying the Ratio Test

To apply the ratio test, consider the terms \( a_n = \frac{(-1)^n}{n+1} \) and \( a_{n+1} = \frac{(-1)^{n+1}}{n+2} \). The ratio is given by\[\left| \frac{a_{n+1} x^{2(n+1)}}{a_n x^{2n}} \right| = \left| \frac{(-1)^{n+1}}{n+2} \times \frac{n+1}{(-1)^n} \times x^2 \right| = \left| \frac{n+1}{n+2} \right| \times |x^2|.\]
03

Simplifying the Ratio

Simplifying the ratio gives\[\lim_{n \to \infty} \left| \frac{n+1}{n+2} \right| \cdot |x^2| = \lim_{n \to \infty} \left(1 - \frac{1}{n+2}\right) \cdot x^2 = |x^2| = |x|^2.\]
04

Determining Convergence from the Ratio Test

For convergence of the series, the limit from the ratio test must be less than 1:\[|x|^2 < 1 \|x| < 1.\]This inequality tells us that the series converges when \( -1 < x < 1 \).
05

Testing the Endpoints

Now, we need to test the endpoints \( x = -1 \) and \( x = 1 \).**At** \( x = -1 \): The series becomes \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(-1)^{2n}}{n+1} = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} \), which converges by the alternating series test.**At** \( x = 1 \): The series becomes \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} \), which also converges by the alternating series test.
06

Concluding the Interval of Convergence

Combining these results, the interval of convergence of the given series is \( [-1, 1] \), including the endpoints since the series converges at both.

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Key Concepts

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

Power Series
A power series is an infinite series of the form \( a_n x^n \), where \( n \) is a non-negative integer, \( a_n \) are coefficients, and \( x \) is a variable. In our given exercise, the series is \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} x^{2n} \). The general formula takes the form \( a_n x^{2n} \) because every term involves \( x \) raised to the power \( 2n \), which is a characteristic of power series: terms grow with successive powers of \( x \).

**Characteristics of a Power Series**:
  • Power series converge at a particular value of \( x \), known as the radius of convergence.
  • The convergence behavior around the center can determine the interval over which the series converges.
  • Often used to represent functions as infinite polynomials.
Understanding the range where the series converges, called the interval of convergence, is a key part of working with power series.
Ratio Test
The ratio test is a method used to determine the convergence of a series. We use it by examining the limit of the ratios between successive terms. In our exercise, the series provided is \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} x^{2n} \).

**Steps using the Ratio Test**:
  • Compute \( a_{n+1} / a_n \). For our series, this was calculated as \( \left| \frac{n+1}{n+2} \right| \times |x^2| \).
  • Find the limit: \( \lim_{n \to \infty} \left(1 - \frac{1}{n+2}\right) \times x^2 = |x|^2 \).
  • For the series to converge, the limit must be less than 1. Thus, \( |x|^2 < 1 \) simplifies to \( |x| < 1 \).
By using the ratio test, we found out that the power series converges for \( |x| < 1 \), giving us the basic interval.
Convergence
Convergence in sequences or series means that they approach a specific value as more terms are added. In the context of this exercise, convergence of the power series determines where the series sums to a finite number.

**Key Points on Convergence**:
  • For power series, convergence requires evaluating limits to find intervals.
  • Using the ratio test helps identify a preliminary range (like \( |x| < 1 \) in our exercise).
  • However, testing endpoints of the interval is crucial, as it confirms whether the series converges or diverges at boundary values (e.g., \( x = -1 \) and \( x = 1 \)).
In our series, after employing the ratio test and testing endpoints, both ends converge, leading to a comfortable interval \( [-1, 1] \).
Alternating Series Test
The alternating series test is a specific convergence test used for series whose terms alternate in sign. Observing the series at the endpoints using this method confirms if the series converges.

**Using the Alternating Series Test**:
  • The series \( \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n+1} \) was assessed at both endpoints in the exercise.
  • The test considers whether the absolute value of the terms decreases as \( n \) increases.
  • The test also requires that as \( n \to \infty \), the terms approach zero.
By testing \( x = -1 \) and \( x = 1 \), both endpoints met the criteria for the alternating series test, confirming convergence and secure endpoint inclusion in the interval of convergence \( [-1, 1] \).

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