Problem 2 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Finden Sie eine zahl \(x \in \mathbb{R}\) so, daB die Reihe \(\sum_{k=1}^{\infty} x^{k-1}\) konvergent ist. Finden sie eine zahl \(y \in \mathbb{R}\) so, \(\operatorname{da} \beta\) die Reihe \(\sum_{k=1}^{\infty} y^{k-1}\) divergent ist.

Step-by-Step Solution

Answer

Choose \( x = 0.5 \) (convergent) and \( y = 2 \) (divergent).

Step 1: Identify Convergence Conditions

The given series is \( \sum_{k=1}^{\infty} x^{k-1} \), which is a geometric series. A geometric series \( \sum_{k=0}^{\infty} ar^k \) converges if the absolute value of the common ratio \( |r| < 1 \). For our series, the general term is \( x^{k-1} \), so we identify \( r = x \). Therefore, the series converges if \( |x| < 1 \).

Step 2: Find a Value of \(x\) for Convergence

To ensure the series \( \sum_{k=1}^{\infty} x^{k-1} \) converges, we choose any \( x \) such that \( |x| < 1 \). A simple choice is \( x = 0.5 \), as \( |0.5| < 1 \).

Step 3: Identify Divergence Conditions

Now consider the series \( \sum_{k=1}^{\infty} y^{k-1} \). For a geometric series to diverge, the requirement is that \( |y| \geq 1 \).

Step 4: Find a Value of \(y\) for Divergence

To ensure the series \( \sum_{k=1}^{\infty} y^{k-1} \) diverges, choose any \( y \) such that \( |y| \geq 1 \). A simple choice is \( y = 2 \), as \( |2| = 2 > 1 \).

Key Concepts

Series Convergence

When we talk about series convergence in the context of a geometric series, we are referring to whether the series equates to a finite number. A geometric series has the general form \( \sum_{k=0}^{\infty} ar^k \), where \( a \) is the initial term and \( r \) is the common ratio between successive terms. For a geometric series to converge, the absolute value of its common ratio \( |r| \) must be strictly less than 1.
This means that the terms of the series get smaller and closer to zero as we progress towards infinity, resulting in a finite sum.

• If \( |r| < 1 \), the series converges.
• If \( x = 0.5 \), for example, since \( |0.5| < 1 \), the series \( \sum_{k=1}^{\infty} 0.5^{k-1} \) will converge.

This translates to the terms shrinking gradually, allowing the sum to reach a stable, finite value.

Series Divergence

On the flip side, series divergence occurs when the series does not settle at a particular value as it extends to infinity. In the case of a geometric series, it diverges if the absolute value of the common ratio \( |r| \) is greater than or equal to 1.

• The terms do not decrease sufficiently, and as a result, the series grows indefinitely or fluctuates without approaching a finite limit.
• For instance, choosing \( y = 2 \), we find \( |2| = 2 \), which means \( |r| \geq 1 \). The geometric series \( \sum_{k=1}^{\infty} 2^{k-1} \) therefore diverges.

This is because the terms grow or remain constant, causing the sum to increase continually with no end in sight.

Real Numbers

Real numbers, denoted by the symbol \( \mathbb{R} \), are an essential concept in mathematics that affects how series behave. Real numbers include all the numbers that can be found on the number line, encompassing both rational numbers (like fractions \( 2/3 \) or integers \( -4 \)) and irrational numbers (like \( \sqrt{2} \) or \( \pi \)).
In the context of the original exercise, we are trying to find a real number \( x \) or \( y \) that makes a geometric series converge or diverge. Whether we are looking for convergence or divergence, we are dealing with numbers that are part of the sets of real numbers.

• This universality allows us to freely select values like \( 0.5 \) or \( 2 \) from \( \mathbb{R} \) to determine the behavior of our series.

Understanding real numbers gives us insight into choosing suitable values that align with the properties of convergence and divergence in geometric series.