Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{(n-1) !} $$

Step-by-Step Solution

Answer

The convergence set is \((-\infty, \infty)\).

Step 1: Identifying the Form of the Series

The given power series is \( \sum_{n=1}^{\infty} \frac{x^{n}}{(n-1)!} \). Notice that, the series has the general term \( a_n = \frac{x^n}{(n-1)!} \). This indicates that the series is a power series centered at 0, given as \( \sum a_n x^n \), where \( a_n = \frac{1}{(n-1)!} \).

Step 2: Applying the Ratio Test

To find the radius of convergence, we will use the Ratio Test. Evaluate \( L = \lim_{n \to \infty} \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| \). Plugging in the terms, we have:\[L = \lim_{n \to \infty} \left| \frac{x^{n+1}}{n!} \cdot \frac{(n-1)!}{x^n} \right| = \lim_{n \to \infty} \left| \frac{x}{n} \right| = 0\]

Step 3: Interpreting L

The limit \( L = 0 \) for all \( x \) means that the series converges absolutely for all values of \( x \). This is because the Ratio Test tells us that if \( L < 1 \), the series converges absolutely.

Step 4: Conclusion on Convergence Set

Since \( L = 0 < 1 \) for all \( x \), the series converges for every real number. Hence, the convergence set of the series is the entire real line, i.e., \( (-\infty, \infty) \).

Key Concepts

Power Series

A power series is a series of the form \( \sum_{n=0}^{\infty} a_n x^n \), where \( x \) is a variable and \( a_n \) are constants known as the coefficients of the series. These series are centered around a particular point; in many cases, this is 0, making it simpler to handle. Power series can be viewed as infinite polynomials that converge within particular intervals. Understanding when these series converge is crucial for applications in calculus and mathematical analysis.

### Characteristics of Power Series
- **Terms**: Each term in the series involves some power of \( x \) multiplied by a coefficient \( a_n \).
- **Center**: The power series is often centered at a point, often 0 (as in our exercise), making it take the form \( a_n x^n \).
- **Radius of Convergence**: This tells us how far from the center the series remains convergent. In simple terms, it is the "distance" within which the series will function as expected.
Understanding the nature of power series is essential as they appear frequently across different areas of mathematics and even in physics applications, modeling phenomena from motion to heat transfer.

Ratio Test

The ratio test is a common method to determine whether a series converges. In terms of a power series, it helps ascertain the radius of convergence, which is the range of \( x \) values for which the series converges. The ratio test involves examining the limit of the ratio of consecutive terms of the series.

### How the Ratio Test Works
- **Set Up**: Consider \( a_n \) as the general term in a series. Compute \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
- **Decision Criteria**: Depending on the value of \( L \): * If \( L < 1 \), the series converges absolutely. * If \( L > 1 \) or \( L = \infty \), the series diverges. * If \( L = 1 \), the test is inconclusive.
In our original exercise, the ratio test found \( L = 0 \) for all \( x \), indicating convergence for all \( x \).

This method is favored because of its simplicity and efficiency, particularly for power series, where ratios simplify calculations involving factorials or exponential terms.

Radius of Convergence

The radius of convergence is a critical concept when dealing with power series. It represents the "radius" of the interval centered around the point of expansion (often 0) within which a power series converges. Determining this radius helps in understanding the behavior of the series across its domain.

### Determining the Radius of Convergence
- The radius of convergence, \( R \), is found using criteria such as the ratio test. For a power series \( \sum_{n=0}^{\infty} a_n x^n \), the formula often used is: \[ \frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]- In cases where the limit evaluates to zero for all \( x \), the radius is infinite, and the series converges everywhere on the real line.

In the exercise, the outcome \( L = 0 \) indicated an infinite radius of convergence, meaning the power series converges for any real number \( x \).

Understanding the radius of convergence provides insights into where a power series functions effectively and helps predict its behavior beyond the immediate area of convergence.