Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{3^{n}} $$

Step-by-Step Solution

Answer

The convergence set for the series is \( (-3, 3) \).

Step 1: Identify the Power Series

The given power series is \( \sum_{n=1}^{\infty} \frac{x^{n}}{3^{n}} \). This series represents a function in terms of \( x \), with coefficients \( a_n = \frac{1}{3^n} \).

Step 2: Recognize the Power Series Form

The standard form of a power series is \( \sum_{n=0}^{\infty} c_n x^n \). Our series can be rewritten in this form as \( \sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n x^n \). This is a geometric series with a common ratio \( r = \frac{x}{3} \).

Step 3: Determine the Ratio of Convergence

For a geometric series \( \sum_{n=0}^{\infty} ar^n \), the series converges if the absolute value of the common ratio is less than 1, \( |r| < 1 \). Here, \(|\frac{x}{3}| < 1\).

Step 4: Solve the Convergence Inequality

Solving \(|\frac{x}{3}| < 1\) gives us \(-1 < \frac{x}{3} < 1\). Multiplying through by 3, we get \(-3 < x < 3\).

Step 5: Conclude the Convergence Set

Hence, the convergence set (interval of convergence) for the power series is \( (-3, 3) \).

Key Concepts

Geometric Series

A geometric series is a series of the form \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio. Each term of the series is a multiple of the previous term by \( r \). This is a simple yet powerful concept in mathematics. Geometric series are significant because they have a straightforward convergence criterion based on the common ratio.

• If \( |r| < 1 \), the series converges.
• If \( |r| \geq 1 \), the series diverges.

Understanding geometric series is crucial when dealing with power series, as many power series can be manipulated into this form to easily determine convergence within certain intervals.

Interval of Convergence

The interval of convergence of a power series is the set of all values \( x \) for which the series converges. For our given series, \( \sum_{n=1}^{\infty} \frac{x^{n}}{3^{n}} \), recognizing it as a geometric series helps us find the interval of convergence. By determining where the common ratio remains within the acceptable range \( |r| < 1 \), we solve:

• \( \left| \frac{x}{3} \right| < 1 \)
• Which simplifies to \( -3 < x < 3 \)

This yields the interval \( (-3, 3) \), meaning within this range, the series will converge. It's important to check endpoints separately to determine if they should be included, but for a simple geometric setup like this, endpoints aren't typically included unless additional context specifies otherwise.

Ratio Test

The ratio test is a powerful tool to determine the convergence of series, especially when terms of the series involve factorials or powers. Although it’s not strictly necessary for geometric series, understanding the ratio test expands our ability to handle more complex power series. The ratio test uses:

• \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)
• If this limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, or infinity, the series diverges.
• If the limit equals 1, the test is inconclusive.

For the series \( \sum \frac{x^n}{3^n} \), applying the ratio test will also reaffirm convergence where \( |\frac{x}{3}| < 1 \), aligning with our geometric series approach.

Absolute Convergence

Absolute convergence is a stronger form of convergence that ensures a series converges even if the terms' signs alternate. A series \( \sum a_n \) is said to converge absolutely if the series of absolute values \( \sum |a_n| \) converges. This concept guarantees convergence even if rearranging terms, which isn't the case for conditional convergence.

• If a series is absolutely convergent, then it is also convergent.
• Absolute convergence implies the overall behavior is stable under various conditions.

For the power series \( \sum \frac{x^n}{3^n} \), proving absolute convergence within the interval \( (-3, 3) \) can be verified using both the geometric series approach and, if necessary, the ratio test, both of which demonstrate stability and convergence for all \( x \) in the interval, thus ensuring reliable series behavior across permissible \( x \) values.