Problem 5 - Basic Analysis: Introduction to Real Analysis

Question

For the following power series, find if they are convergent or not, and if so find their radius of convergence. a) \(\sum_{n=0}^{\infty} 2^{n} x^{n}\) b) \(\sum_{n=0}^{\infty} n x^{n}\) c) \(\sum_{n=0}^{\infty} n ! x^{n}\) d) \(\sum_{n=0}^{\infty} \frac{1}{(2 n) !}(x-10)^{n}\) e) \(\left.\sum_{n=0}^{\infty} x^{2 n} \quad f\right) \sum_{n=0}^{\infty} n ! x^{n !}\)

Step-by-Step Solution

Answer

a) R=0.5; b) R=1; c) R=0; d) R=∞; e) R=1; f) R=0. Convergence depends on the radius for each.

Step 1: Analyze Series a

To determine the radius of convergence of the series \( \sum_{n=0}^{\infty} 2^n x^n \), we apply the formula for the radius of convergence: \[ R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}} \]Here, \( a_n = 2^n \). Thus, \[ \sqrt[n]{|a_n|} = \sqrt[n]{2^n} = 2 \]Therefore, \( \limsup_{n \to \infty} \sqrt[n]{|a_n|} = 2 \), and hence:\[ R = \frac{1}{2} \] The series converges for \( |x| < \frac{1}{2} \).

Step 2: Analyze Series b

The series \( \sum_{n=0}^{\infty} n x^n \) can be evaluated using the radius of convergence formula. Here, \( a_n = n \), so:\[ \limsup_{n \to \infty} \sqrt[n]{n} = 1 \] Thus, the radius is given by:\[ R = \frac{1}{1} = 1 \]The series converges for \( |x| < 1 \).

Step 3: Analyze Series c

For the series \( \sum_{n=0}^{\infty} n! x^n \), calculate the radius of convergence:\[ \limsup_{n \to \infty} \sqrt[n]{n!} \to \infty \] because factorials grow very quickly, leading to:\[ R = \frac{1}{\infty} = 0 \]This series converges only at \( x = 0 \).

Step 4: Analyze Series d

Consider \( \sum_{n=0}^{\infty} \frac{1}{(2n)!}(x-10)^n \): \( a_n = \frac{1}{(2n)!} \), and \[ \limsup_{n \to \infty} \sqrt[n]{\frac{1}{(2n)!}} = 0 \] because \( (2n)! \) grows extremely fast, giving us:\[ R = \infty \]The series converges for all \( x \).

Step 5: Analyze Series e

For \( \sum_{n=0}^{\infty} x^{2n} \), recognize this as a geometric series with \( a = 1 \) and \( r = x^2 \), which converges when \( |x^2| < 1 \) or \( |x| < 1 \). Thus, the radius is:\[ R = 1 \]

Step 6: Analyze Series f

For \( \sum_{n=0}^{\infty} n! x^{n!} \), using the same reasoning as in series c, the terms grow quickly:\[ \limsup_{n \to \infty} \sqrt[n]{n!} \to \infty \]Thus, the radius of convergence is:\[ R = 0 \]The series converges only at \( x = 0 \).

Key Concepts

Power Series

A power series is a series of the form \( \sum_{n=0}^{\infty} a_n x^n \), where \( a_n \) is the coefficient of each term and \( x \) is a variable. Understanding power series is fundamental to many areas in mathematics, particularly calculus and complex analysis.
A power series converges when the variable \( x \) lies within a particular region around a central point. This region is determined by the radius of convergence, which defines how far from the center point the series will converge.
When working with power series, it's essential to understand:

• Convergence: Whether the series sums up to a finite value in a particular range of \( x \)
• Radius of Convergence \( R \): Indicates the interval \( |x| < R \) where the series converges

The radius of convergence can be found using various methods, such as the Ratio Test or Limsup Method. These methods help quantify the growth of terms in the series to determine where the series behaves nicely and sums up correctly.

Factorial

The factorial of a non-negative integer \( n \), denoted \( n! \), is the product of all positive integers less than or equal to \( n \). Factorials play a crucial role in combinations, permutations, and series such as Taylor series.
A crucial point about factorials is their rapid growth rate. For example:

• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)

This rapid growth is significant when evaluating series with factorial terms, as it often leads to infinite radii of convergence, or in some cases, the series may converge only at a specific value of \( x \), like when the series contains \( n! x^n \).

Geometric Series

A geometric series is a series with a constant ratio between successive terms. It takes the form \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio.
Geometric series are a fundamental concept and are one of the few infinite series that possess a simple convergence condition:

• The series converges when the absolute value of the ratio \( |r| < 1 \)
• The sum of a convergent geometric series is given by \( \frac{a}{1-r} \)

This simplicity makes geometric series a powerful tool for understanding more complicated series and serves as a gateway into the study of power series. When a power series can be simplified to a geometric form, it immediately provides insights into the radius of convergence.

Limsup

The term \( \limsup \), or limit superior, is a concept used in analysis that describes the limiting behavior of a sequence. In the context of series, it helps determine the radius of convergence by evaluating the growth of the series terms.
For a sequence \( \{a_n\} \), the \( \limsup \) of \( a_n \) as \( n \to \infty \) is the greatest limit point of the sequence. It's a way to capture the "upper bound trend" of the sequence. In terms of finding the radius of convergence \( R \), it is used in:

• The formula \( R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|a_n|}} \)

This formula provides a systematic way to determine \( R \) by evaluating the speed at which the terms \( \{a_n\} \) grow, or shrink, as \( n \) increases. By understanding \( \limsup \), mathematicians and students can assess the convergence nature of complex power series effectively.