Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the radius of convergence of $$ \sum_{n=0}^{\infty} \frac{(p n) !}{(n !)^{p}} x^{n} $$ where \(p\) is a positive integer.

Step-by-Step Solution

Answer

The radius of convergence is \( \frac{1}{p^p} \).

Step 1: Apply the Ratio Test

To find the radius of convergence, we use the ratio test, which considers the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \), where \( a_n = \frac{(p n)!}{(n!)^p} x^n \). Compute the ratio of consecutive terms:\[\frac{a_{n+1}}{a_n} = \frac{\frac{(p(n+1))!}{((n+1)!)^p} x^{n+1}}{\frac{(pn)!}{(n!)^p} x^n} = \frac{(p(n+1))!}{(pn)!} \cdot \left(\frac{n!}{(n+1)!}\right)^p \cdot x\]

Step 2: Simplify the Factorials

Simplify the expression for \( \frac{(p(n+1))!}{(pn)!} \) and combine it with the second factor:\[\frac{a_{n+1}}{a_n} = \frac{(p(n+1))(p(n+1)-1)\cdots (pn+1)}{(n+1)^p} x\]As \(n\) approaches infinity, the leading behavior of the term \((p(n+1))! / (pn)! \approx (pn)^p\), allowing simplification to:\[\frac{a_{n+1}}{a_n} \approx p^p x\]

Step 3: Find the Limit Using the Ratio Test

The ratio test tells us to evaluate the limit \( L = p^p |x| \). For convergence, we need \( L < 1 \). This implies:\[p^p |x| < 1 \Longrightarrow |x| < \frac{1}{p^p}\]

Step 4: Determine the Radius of Convergence

The radius of convergence \( R \) is the value of \( |x| \) such that the series converges. From the inequality from the previous step, we have:\[R = \frac{1}{p^p}\]This is the radius of convergence for the given power series.

Key Concepts

Power Series

Power series are infinite series of the form \( \sum_{n=0}^{\infty} a_n x^n \). They are expressions for functions as an infinite sum of terms. Think of a power series as an endless polynomial. Each term involves a coefficient \(a_n\) and a variable raised to a power. In this context, our power series is given by:\[ \sum_{n=0}^{\infty} \frac{(p n) !}{(n !)^{p}} x^{n} \]This specific series involves factorials in both the numerator and the denominator. The coefficients in the series determine how quickly the series converges or diverges. Finding where these infinite sums converge is crucial, which leads us to discuss the radius of convergence.

Ratio Test

The ratio test is a method used to determine the convergence of a series. It involves taking a limit of the absolute value of the ratio of consecutive terms. For a series \( \sum a_n \), the ratio test considers:\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]If \( L < 1 \), the series converges absolutely. If \( L > 1 \), the series diverges. If \( L = 1 \), the test is inconclusive. In our problem, the ratio \( \frac{a_{n+1}}{a_n} \) calculates the relationship between consecutive terms, helping us determine the values of \( x \) for which the series will converge.

Factorial Simplification

Factorial simplification is crucial when dealing with expressions that contain factorials, especially when applying the ratio test. Factorials grow very quickly, so simplifying them helps us understand the behavior of large expressions. In this exercise, a factorial expression is given:\[ \frac{(p(n+1))!}{(pn)!} \cdot \left(\frac{n!}{(n+1)!}\right)^p \]To simplify, notice that the first part expands, and thus we can cancel terms to simplify the fraction:- Remove common terms in the numerator and denominator.- Approximate using dominant terms as \( n \to \infty \).The simplification reveals that the main influence on the convergence comes from the factor \( p^p \) multiplying \( x \). This leads to easier computation of the limit.

Limit Evaluation

Limit evaluation involves finding the limit of a simplified expression to determine convergence. After simplifying the ratio of terms:\[ \frac{a_{n+1}}{a_n} \approx p^p x \]As we apply the ratio test, we find the limit:\[ L = \lim_{n \to \infty} p^p |x| \]For the series to converge, \( L \) must be less than 1. This gives us the condition:\[ p^p |x| < 1 \]Solving for \( |x| \), this inequality shows us that the series converges for values:\[ |x| < \frac{1}{p^p} \]Thus, the radius of convergence \( R \) is \( \frac{1}{p^p} \). Evaluating limits is a key step in pinpointing where exactly the power series will converge.