Problem 27 - Calculus: Early Transcendentals

Question

Find the radius of convergence and interval of convergence of the series. \( \sum_{n = 1}^{\infty} \frac {x^n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot\cdot \cdot (2n - 1)} \)

Step-by-Step Solution

Answer

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

Step 1: Identify the General Term

The general term of the given series is \( a_n = \frac{x^n}{(2n-1)!!} \), where \((2n-1)!!\) denotes the double factorial of odd numbers up to \((2n-1)\).

Step 2: Apply the Ratio Test for Convergence

To find the radius of convergence, apply the ratio test. Calculate \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{x^{n+1}}{(2n+1)!!} \cdot \frac{(2n-1)!!}{x^n} \right| = \lim_{n \to \infty} \left| \frac{x}{2n+1} \right| \).

Step 3: Simplify the Limit

Evaluate the simplified form: Since \( \frac{x}{2n+1} \to 0 \) as \( n \to \infty \) for any finite \( x \), the limit is zero.

Step 4: Conclude Radius of Convergence

Since the limit from the ratio test equals zero for any \( x \), the series converges for all \( x \). Therefore, the radius of convergence is infinite.

Step 5: Determine the Interval of Convergence

The interval of convergence is determined by the radius of convergence. Since the radius is infinite, the interval of convergence is \((-\infty, \infty)\).

Key Concepts

Interval of Convergence

The interval of convergence refers to the set of all values of \( x \) for which a given series converges. To find this interval, it is crucial to determine where the series actually converges. Once we know the radius of convergence—let's call it \( R \)—we can often say that the interval of convergence is within \( (a - R, a + R) \), where \( a \) is the center of the power series.In simpler terms, the interval of convergence tells us the "comfortable zone" for \( x \), in which the series behaves nicely and doesn't smash to infinity. In the given problem, since the radius of convergence is infinite, it means our interval stretches from \(-\infty\) to \(\infty\). This tells us that the series converges for every real number \( x \). So, unlike typical series which have a smaller interval, this one behaves for every single value of \( x \).

Ratio Test

The ratio test is a handy tool for determining the convergence of a series. It's based on comparing the size of consecutive terms in a sequence. Here's how it works: if the ratio of successive terms \( \left| \frac{a_{n+1}}{a_n} \right| \) tends toward a value less than 1 as \( n \) becomes very large, the series converges absolutely.To apply the ratio test:

• Compute \( \left| \frac{a_{n+1}}{a_n} \right| \)
• Take the limit as \( n \to \infty \)
• Check the result

In your exercise, the ratio test confirms convergence for any \( x \) because the limit equals zero. This implies a radius of convergence that is infinite, leading to an interval of convergence that spans all real numbers. It's like a magical pass that lets you know the series is friendly over any number line!

Double Factorial

The double factorial is a lesser-known operation, but it's quite crucial for problems like this. Notated as \((2n-1)!!\), the double factorial of an odd number is the product of all odd numbers up to that odd number. For instance, \((5)!! = 5 \cdot 3 \cdot 1\).Imagine it this way: you skip every second number from the sequence of positive integers, then multiply the remaining numbers! In this problem, \((2n-1)!!\) stands for the product all odd numbers up to \((2n-1)\).This is important because the series term involves this special kind of factorial, making the series decrease fast and helps it converge on a wider set of \( x \) values. The double factorial makes the denominator grow significantly large, which makes the whole term smaller.

Infinite Convergence

Infinite convergence is a phenomenon where a series converges regardless of how large or small the values of \( x \) are. It's a pretty big deal in the world of mathematics because it means that no matter what value you substitute for \( x \), the series will behave nicely and converge.For the series in the exercise, the conclusion that the series converges for all \( x \) comes from the fact that applying the ratio test led to a limit of zero. Essentially, this means there's no real number \( x \) that would cause the series to diverge, hence an infinite radius of convergence.Think of it as a never-ending friendship between \( x \) and the series—they simply get along no matter the situation. It's a reassuring property because mathematicians and enthusiasts can experiment with \( x \) freely without worrying about the series going out of bounds.