Problem 4 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Uberprüfen Sie noch einmal mit der Formel von Hadamard, für welche \(x \in \mathbb{R}\) die geometrische Reihe $$ \begin{gathered} \sum_{n=O}^{\infty} x^{n} \\ \text { konvergiert. } \end{gathered} $$

Step-by-Step Solution

Answer

The series converges for \( x \in (-1, 1) \).

Step 1: Identify the Geometric Series

The given series is a geometric series of the form \( \sum_{n=0}^{\infty} x^n \). To determine the convergence of this series, we need to apply the formula for a geometric series.

Step 2: Recall the Condition for Convergence of Geometric Series

A geometric series \( \sum_{n=0}^{\infty} ar^n \) converges if the absolute value of the common ratio \( |r| < 1 \). In our series, \( r = x \), so we need to find for which \( x \) the condition \( |x| < 1 \) holds.

Step 3: Apply Hadamard's Formula

Hadamard's formula for the convergence of power series states that the series converges when \( |x| < R \), where \( R \) is the radius of convergence given by \( R = 1/L \), with \( L = \limsup_{n \to \infty} |a_n|^{1/n} \). For our series, \( a_n = x^n \), making \( |a_n|^{1/n} = |x| \).

Step 4: Determine the Radius of Convergence

Compute \( L = \limsup_{n \to \infty} |x^n|^{1/n} = |x| \). Thus, according to Hadamard's formula \( R = 1/|x| \), thus the series converges when \( |x| < 1 \).

Step 5: State the Interval of Convergence

Thus, the geometric series \( \sum_{n=0}^{\infty} x^n \) converges for \( x \in (-1, 1) \).

Key Concepts

Convergence

Convergence in the context of series is all about whether the sum of an infinite sequence of terms approaches a particular value. For geometric series, this concept translates into understanding when and why the sum does not grow infinitely large but settles around a finite number as more terms are added.
One important thing to remember is that a geometric series will converge if the common ratio, denoted by \( r \), has an absolute value less than one. This condition ensures that each subsequent term becomes progressively smaller and influences the sum less significantly as the series progresses.
When discussing convergence, especially within a geometric series, you make sure the terms shrink enough so that their total does not escalate without bounds. Hence, for a geometric series like \( \sum_{n=0}^{\infty} x^n \), it converges when \( |x| < 1 \), ensuring the terms of the series add up to a manageable, finite value.

Radius of Convergence

The radius of convergence is a key concept used in analyzing power series. It reveals how far you can "stretch" the series before it stops converging. Imagine it as a safe zone where we can use the series to get reliable results without worry.
In the case of the geometric series \( \sum_{n=0}^{\infty} x^n \), the radius of convergence is derived from the nature of the common ratio, \( x \). Mathematically, this radius is the value \( R \) for which the series converges for all absolute values \( |x| < R \).
Using mathematical tools such as Hadamard’s formula, we calculate the radius of convergence. For this geometric series, the result is \( |x| < 1 \). Essentially, within this radius, the series behaves nicely and converges, giving us confidence in its stability and reliability.

Hadamard's Formula

Hadamard's formula is a brilliant mathematical tool for finding the convergence of power series. It simplifies the process by providing a method to determine the radius of convergence for a given series.
This formula specifies that the radius of convergence \( R \) is given by \( R = 1/L \), where \( L \) is the limit superior, or \( \limsup \), of the sequence\( |a_n|^{1/n} \). Essentially, it captures the most impactful factor influencing the growth of series terms.
For our geometric series \( \sum_{n=0}^{\infty} x^n \), the application of Hadamard’s formula confirms that \( R = 1/|x| \), leading us to validate that the series converges when \( |x| < 1 \).
So, Hadamard's formula helps quantify how trustworthy the series remains over different values, ensuring it only converges within the prescribed boundary.