Problem 14 - Calculus: Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {x^{2n}}{n!} \)

Step-by-Step Solution

Answer

The radius of convergence is infinite, and the interval is \((-\infty, \infty)\).

Step 1: Identify the General Term

The given series is \( \sum_{n=1}^{\infty} \frac{x^{2n}}{n!} \). The general term of this series is \( a_n = \frac{x^{2n}}{n!} \).

Step 2: Apply the Ratio Test

To find the radius of convergence, we use the Ratio Test. Compute the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \):\[\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{x^{2(n+1)}}{(n+1)!} \cdot \frac{n!}{x^{2n}} \right| = \left| \frac{x^2 \cdot x^{2n}}{x^{2n} \cdot (n+1)} \right| = \left| \frac{x^2}{n+1} \right|.\]

Step 3: Find the Limit of the Ratio

Now find the limit as \( n \to \infty \):\[\lim_{n \to \infty} \left| \frac{x^2}{n+1} \right| = \lim_{n \to \infty} \frac{|x|^2}{n+1} = 0.\]This limit is always 0 for any value of \( x \), hence it is less than 1 for all \( x \).

Step 4: Determine the Radius of Convergence

Since the Ratio Test results in a limit of 0, which is less than 1, the radius of convergence \( R \) is infinite. This means the series converges for all real numbers \( x \).

Step 5: State the Interval of Convergence

The interval of convergence is \( (-\infty, \infty) \) due to the infinite radius of convergence. This implies the series converges for all real values of \( x \).

Key Concepts

Interval of Convergence

The interval of convergence refers to the set of all real numbers for which a given infinite series converges. It tells us the "span" or "range" of values for which the series sums up to a finite number.
Understanding the interval of convergence is crucial when expanding functions into power series. It determines where these functions can be accurately represented by the series.

• For example, if a series converges for all real numbers, its interval of convergence is \((-\infty, \infty)\).
• If a series only converges for values between -3 and 3, its interval is \((-3, 3)\).

In this exercise, the interval of convergence was found to be \((-\infty, \infty)\), meaning the series converges for any real value of \ x \.

Ratio Test

The Ratio Test is a method to determine the convergence or divergence of an infinite series. It involves the calculation of the limit of the absolute value of the ratio of consecutive terms in the series.

• For a series \( \sum_{n=1}^{\infty} a_n \), consider the terms \( a_n \) and \( a_{n+1} \).
• The ratio test computes \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

Applications of the Ratio Test

• If the limit is less than 1, the series converges absolutely.
• If the limit is greater than 1, or if the limit does not exist, the series diverges.
• If the limit equals 1, the test is inconclusive.

In the given exercise, applying the Ratio Test yields a limit of 0, which is less than 1, indicating absolute convergence for any \ x \.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. It can be expressed as \( \sum_{n=1}^{\infty} a_n \), where each \( a_n \) represents a term in the series.
Infinite series appear frequently in mathematics and have applications in various fields such as calculus, physics, and engineering.
There are several types of infinite series:

• Geometric series, where each term is a constant multiple of the previous one.
• Arithmetic series, with a constant difference between successive terms.
• Power series, expressed in terms of powers of a variable, like \( x \).

In this context, we're examining a power series of the form \( \sum_{n=1}^{\infty} \frac{x^{2n}}{n!} \), which uses factorials in the denominator.

Convergence of Series

The convergence of a series refers to whether the sum of an infinite sequence pulls towards a finite value. Convergence is a key concept that tells us if we can make sense of the sum of infinitely many terms.

Conditions for Convergence

• A series converges if the sequence of partial sums approaches a finite limit.
• If the partial sums continue increasing or decreasing without bound, the series is said to diverge.

In this particular exercise, it's concluded that the series converges for all values of \( x \), due to the infinite radius of convergence. This means, no matter what real number you plug into the series, it will add up to a finite sum, likely bringing stability and predictability in practical applications.