Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 32
Find the radius of convergence of $$ \sum_{n=0}^{\infty} \frac{(p n) !}{(n !)^{p}} x^{n} $$ where \(p\) is a positive integer.
Short Answer
Expert verified
The radius of convergence is \( R = \frac{1}{p^p} \).
Step by step solution
01
Identify the Ratio Test Formula
To find the radius of convergence, we'll use the Ratio Test. The Ratio Test states that a series \( \sum a_n \) converges absolutely if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \).
02
Determine the General Term
The general term of the series is \( a_n = \frac{(p n)!}{(n!)^p} x^n \). We need to evaluate the ratio \( \frac{a_{n+1}}{a_n} \).
03
Set Up the Ratio
Compute \( \frac{a_{n+1}}{a_n} = \frac{(p(n+1))!}{((n+1)!)^p} x^{n+1} \times \frac{(n!)^p}{(pn)! x^n} \).
04
Simplify the Ratio
Simplify \( \frac{a_{n+1}}{a_n} = \frac{(p(n+1))!}{(pn)!} \times \frac{(n!)^p}{((n+1)!)^p} \cdot x \).
05
Simplify Further Using Factorials
For the factorials, \( (p(n+1))! = (pn+p)! = (pn+p)(pn+p-1)...(pn+1)(pn)! \). This simplifies the ratio to \( \frac{(pn+1)(pn+2)...(pn+p)}{(n+1)^p} \cdot x \).
06
Take the Limit
Compute \( \lim_{n \to \infty} \left| \frac{(pn+1)(pn+2)...(pn+p)}{(n+1)^p} x \right| \). As \( n \to \infty \), the expression approximates to \( p^p x \).
07
Find the Radius of Convergence
Set \( \left| p^p x \right| < 1 \). Therefore, \( |x| < \frac{1}{p^p} \), leading to the radius of convergence \( R = \frac{1}{p^p} \).
08
Conclusion
The radius of convergence for the series is determined by the condition \( |x| < \frac{1}{p^p} \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Ratio Test
The Ratio Test is a crucial tool in determining whether an infinite series converges. It provides a simple method to evaluate the series by examining the limit of the ratio of consecutive terms. In simpler terms, if you take a series \( \sum a_n \), the Ratio Test says it converges absolutely if the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \).
This means that as you progress along the series, the amount by which each term differs from the previous one diminishes sufficiently over time.
Here are a few key points about the Ratio Test:
This means that as you progress along the series, the amount by which each term differs from the previous one diminishes sufficiently over time.
Here are a few key points about the Ratio Test:
- It efficiently checks for absolute convergence, which is stronger than just convergence.
- If the ratio \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1 \), then the test is inconclusive.
- It is particularly useful for series involving factorials or exponential functions.
Diving into Power Series
A power series is a series of the form \( \sum_{n=0}^{\infty} a_n x^n \), where each term is a power of \( x \) multiplied by a coefficient \( a_n \). Power series are fundamental in mathematical analysis, especially in functions and calculus.
They allow us to represent functions in terms of an infinite sum of powers of \( x \). One might consider them as an extension of polynomials with infinite terms.
Key characteristics of a power series:
They allow us to represent functions in terms of an infinite sum of powers of \( x \). One might consider them as an extension of polynomials with infinite terms.
Key characteristics of a power series:
- The coefficients \( a_n \) determine the series' behavior.
- The variable \( x \) is what you're plugging into the function.
- They have a radius of convergence, which is the distance from the center point \( x = 0 \) within which the series converges.
The Role of Factorials
Factorials are mathematical expressions symbolized by \( n! \), representing the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). They appear frequently in series, especially in problems dealing with convergence or permutations.
In the context of the problem, the expression \( (pn)!/(n!)^p \) encapsulates the use of factorials.
Here's why factorials are significant:
In the context of the problem, the expression \( (pn)!/(n!)^p \) encapsulates the use of factorials.
Here's why factorials are significant:
- They grow very quickly, which can influence the convergence of a series.
- Manipulating factorials requires understanding their properties, like dividing or simplifying terms.
- They offer a natural way to describe permutations and combinations.
Deep Dive into Convergence Analysis
Convergence analysis is the study of whether an infinite series approaches a finite limit. It's a key aspect when dealing with series, especially in calculus and analysis.
This concept addresses whether the terms of a series get closer to a particular value or not.
Factors affecting convergence:
This concept addresses whether the terms of a series get closer to a particular value or not.
Factors affecting convergence:
- The size and pattern of the series' terms, which can be evaluated using tests like the Ratio Test.
- The presence of factorials or other rapidly increasing terms.
- The interplay between terms, like \( a_n x^n \) in a power series.
