Problem 7 - Calculus: Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 0}^{\infty} \frac {x^n}{n!} \)

Step-by-Step Solution

Answer

Radius of convergence is infinity, interval is \((-\infty, \infty)\).

Step 1: Identify the Series Type

The given series is a power series of the form \( \sum_{n=0}^{\infty} a_n x^n \), where \( a_n = \frac{1}{n!} \). It is necessary to find the radius of convergence for this series.

Step 2: Apply the Ratio Test

To determine the radius of convergence, we can apply the Ratio Test. The Ratio Test involves finding \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). For this series, that would be \( \frac{\frac{1}{(n+1)!}}{\frac{1}{n!}} = \frac{1}{n+1} \).

Step 3: Calculate the Limit

We need to compute the limit of the expression from Step 2. Thus, \( \lim_{n \to \infty} \left| \frac{1}{n+1} \right| = 0 \).

Step 4: Determine the Radius of Convergence

According to the Ratio Test, if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \), the series converges for \( |x| < \frac{1}{L} \). Since \( L = 0 \), the series converges for all real numbers \( x \). Therefore, the radius of convergence is infinity.

Step 5: Identify the Interval of Convergence

Because the series has an infinite radius of convergence, the interval of convergence is all real numbers \( (-\infty, \infty) \).

Key Concepts

Power Series

A power series is a series of the form \( \sum_{n=0}^{\infty} a_n x^n \) where \( a_n \) are coefficients and \( x \) is the variable of the series. Power series play a significant role in mathematical analysis because they can represent functions as infinite sums.
Power series are quite flexible as they converge within a certain interval, known as the interval of convergence. The behavior of these series greatly depends on the values of \( x \).
Here, the given series \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \) is a classic example of a power series, with the general coefficient term \( a_n = \frac{1}{n!} \). Understanding how to work with power series is essential for solving many mathematical problems.

Ratio Test

The Ratio Test is a common method used to determine the convergence of a power series. It involves the calculation of the limit of the ratio of successive terms of the series. By finding this limit, you can determine the radius of convergence—a crucial step in analyzing a power series.
The Ratio Test formula is given by:

• Find \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

For our specific series, \( a_n = \frac{1}{n!} \), the Ratio Test involves the calculation of \( \frac{1}{n+1} \). This straightforward calculation helps quickly determine the behavior of the series as \( n \to \infty \).

Limit Calculation

Performing the limit calculation is a critical step in the Ratio Test. Once the ratio of successive terms is established, calculating the limit as \( n \) approaches infinity allows us to apply the results of the Ratio Test.
In our power series, the limit calculation involves:

• Finding \( \lim_{n \to \infty} \left| \frac{1}{n+1} \right| \).
• This simplifies to 0, as \( n+1 \) grows infinitely large, making the whole fraction approach 0.

When this limit is zero, it indicates that the series converges for all real numbers \( x \). This crucial calculation paves the way to establishing both the radius and interval of convergence for the series.

Interval of Convergence

The interval of convergence refers to the set of all \( x \)-values for which the power series converges. Once the radius of convergence is established, determining the interval is straightforward.
For the series \( \sum_{n=0}^{\infty} \frac{x^n}{n!} \), we've discovered that the radius of convergence is infinite, as \( L = 0 \). This tells us:

• The series converges for any real number \( x \), resulting in an infinite interval.
• The interval of convergence is therefore \((-\infty, \infty)\).

Understanding the interval of convergence provides insight into how far the series representation is valid or useful for its related function. This knowledge is essential in both pure and applied mathematics contexts.