Problem 34 - Essential Calculus

Question

In Exercises \(31-34,\) find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \sum_{n=1} \frac{n !(x-c)^{n}}{1 \cdot 3 \cdot 5 \cdots(2 n-1)} $$

Step-by-Step Solution

Answer

The radius of convergence for the power series is infinite. Therefore, the interval of convergence is \((- \infty, + \infty)\). As the radius of convergence is infinite, there are no finite endpoints to check for convergence.

Step 1: Apply the Ratio Test

First, apply the ratio test, which involves taking the limit as \(n\) approaches infinity of the absolute value of the \((n+1)\)th term divided by the \(n\)th term of the series. If this limit, let's call it \(L\), is less than 1, then the series converges.

Step 2: Simplify the expression

After applying the Ratio Test, the expression will be simplified. The term \((x - c)\) will be factored out, and using the properties of factorials, the expression will be simplified to \(L = \frac{(n+1)(x - c)}{2n+1}\).

Step 3: Evaluate the limit

Evaluate the limit of \(L\) as \(n\) approaches infinity. If \(x - c\) is a fixed number, as \(n\) goes to infinity, this limit will equal zero, indicating that the series converges for all \(x\). Hence, the radius of convergence (\(R\)) is infinite.

Step 4: Check the end points

The final step in finding the interval of convergence is to check the endpoints of the interval. However, as the radius of convergence was found to be infinite, there are no finite endpoints to check in this case.

Key Concepts

Power Series

A power series is a type of mathematical series that is particularly useful in areas such as physics and engineering. It takes the form of \( \sum_{n=0}^\infty a_n(x-c)^n \), where \( a_n \) represents the coefficient of the nth term, \( c \) is the center of the series, and \( x \) is the variable. The series is an infinite sum, meaning that it keeps adding terms indefinitely. Think of it like a polynomial with an endless number of terms!

Power series can be used to approximate complicated functions by using polynomials of increasing degree. One of their most remarkable properties is that they can converge (add up to a finite number) within a certain interval around the center \( c \). This interval is where the values of \( x \) lie for which the series is a valid representation of the function. Understanding the notion of convergence is essential for working with power series because it tells us where the series can be used reliably to represent a function.

Ratio Test

The Ratio Test is a powerful tool for determining the convergence of series, especially when dealing with power series. Let's say you have a series \( \sum a_n \), and you want to know if it converges or diverges. The Ratio Test involves calculating the limit of the ratio of the absolute values of consecutive terms, formally written as \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

If the limit \( L \) is less than 1, the series converges absolutely; if \( L \) is greater than 1, it diverges; and if \( L \) equals 1, the test is inconclusive. In the case of a power series, this test helps us determine not just if the series converges, but also the range of values of \( x \) for which it does—fundamentally helping to identify the interval of convergence.

Radius of Convergence

The radius of convergence is a measure that quantifies the distance from the center \( c \) of a power series within which the series converges. It can be found using various tests, with the Ratio Test being one of the most common methods. The result from the Ratio Test gives us the radius \( R \) of the interval \( (c-R, c+R) \) where the series converges.

In cases where the limit \( L \) as \( n \) approaches infinity of the ratio of consecutive terms is zero, we say the radius of convergence is infinite, which means that the series converges for all \( x \) values. However, an infinite radius does not automatically mean that every individual point within that range will converge—endpoints must be checked for convergence. On the other side, if the power series has a finite radius of convergence, it becomes crucial to check the interval's endpoints to determine the series' interval of convergence fully.