Problem 3 - Essential Calculus Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty}(-1)^{n} n x^{n}$$

Step-by-Step Solution

Answer

Radius of convergence is 1; interval of convergence is \((-1, 1)\).

Step 1: Recognize the Series

The given series is \( \sum_{n=1}^{\infty} (-1)^{n} n x^{n} \). This is an alternating power series in \( x \) with general term \( a_n(x) = (-1)^n n x^n \).

Step 2: Apply the Ratio Test

To find the radius of convergence, apply the Ratio Test. Identify the general term as \( a_n = n x^n \). The test requires evaluating \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

Step 3: Evaluate the Limit

Calculate \( \frac{a_{n+1}}{a_n} = \frac{(n+1) x^{n+1}}{n x^n} = \left(1 + \frac{1}{n}\right) |x| \). Taking the limit as \( n \to \infty \), we get \( \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) |x| = |x| \).

Step 4: Determine the Radius of Convergence

By the Ratio Test, the series converges if \( |x| < 1 \). Therefore, the radius of convergence is 1.

Step 5: Test Endpoints for the Interval of Convergence

The radius of convergence tells us that the interval at least goes from \(-1, 1\). We need to test convergence at the endpoints \( x = -1 \) and \( x = 1 \).

Step 6: Test Convergence at \( x = 1 \)

Substitute \( x = 1 \) into the series: \( \sum_{n=1}^\infty (-1)^n n \). This series is equivalent to \( -1 - 2 - 3 - \ldots \), which diverges.

Step 7: Test Convergence at \( x = -1 \)

Substitute \( x = -1 \) into the series: \( \sum_{n=1}^\infty (-1)^n n (-1)^n = \sum_{n=1}^\infty n \), which also diverges.

Step 8: Conclude the Interval of Convergence

Since the series diverges at both endpoints, the interval of convergence for the series is \((-1, 1)\).

Key Concepts

Interval of Convergence

The interval of convergence of a series is the range of values for which the series converges to a sum. For a power series, the interval is found using the radius of convergence and evaluating the behavior at the endpoints. When we have a radius of convergence, the series will converge for all values of the variable within this distance from the center of the series.
In this example, we have determined the radius of convergence to be 1. This tells us that the series converges for values of \( x \) such that \( |x| < 1 \), which means our series starts by converging over the interval from \(-1\) to \(1\) before considering the endpoints.
The endpoints need to be tested separately because sometimes the series converges at one or both ends. Here, we plugged in \( x = 1 \) and \( x = -1 \) into the series and found it diverged in both cases, so neither endpoint is included. Thus, the interval of convergence is \((-1, 1)\), an open interval that does not include the endpoints.

Ratio Test

The ratio test is a method to test the absolute convergence of an infinite series. It involves taking the limit of the absolute value of the ratio of successive terms. For a series \( \sum a_n\), it suggests that if the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \) then:

• The series converges absolutely if \( L < 1 \)
• The series diverges if \( L > 1 \) or \( L \) is undefined
• The test is inconclusive if \( L = 1 \)

In our series, \( a_n = n x^n \), applying the ratio test involves computing \( \frac{a_{n+1}}{a_n} = \left(1 + \frac{1}{n}\right) |x| \). Taking the limit as \( n \) approaches infinity, we simplify to \( |x| \).
Since \( L = |x| \), for the series to converge, we need \( |x| < 1 \). Therefore, the radius of convergence is 1. This radius gives us a crucial insight into where the series will sum up to a finite number and is used to determine the interval of convergence.

Alternating Series

An alternating series is one in which the terms alternate in sign. This means each successive term alternates between positive and negative. For a series like \( \sum (-1)^n a_n \), the terms \(-a_n, a_{n+1} \) switch their signs because of the \((-1)^n\) term.
The series in our problem, \( \sum (-1)^n n x^n \), is an example of an alternating series due to the \((-1)^n\) factor, resulting in alternating positive and negative terms. The alternating pattern plays a key role in determining convergence:

• A series \( \sum (-1)^n b_n \) converges if the terms \( b_n \) are decreasing and \( \lim_{n \to \infty} b_n = 0 \).
• This is known as the Alternating Series Test.

Despite being alternating, our series does not pass the Alternating Series Test at the endpoints \( x = 1 \) and \( x = -1 \), as it diverges. Nonetheless, understanding how alternating series can influence convergence is crucial when dealing with series compositions like these.