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

Question

Konvergieren die folgenden Reihen? Was ist ihre summe? a) \(\sum_{k=1}^{\infty} \frac{2}{3^{k-1}}\) b) \(\sum_{k=1}^{\infty} \frac{2^{k}+3^{k}}{6^{k}}\).

Step-by-Step Solution

Answer

a) Converges, sum = 3. b) Converges, sum = \( \frac{3}{2} \).

Step 1: Recognize the Series Type

For part a), the series is \( \sum_{k=1}^{\infty} \frac{2}{3^{k-1}} \). This suggests a geometric series, identifiable by the constant ratio between successive terms. For part b), \( \sum_{k=1}^{\infty} \frac{2^{k}+3^{k}}{6^{k}} \), notice it's a combination of two geometric series.

Step 2: Calculate the Common Ratio (a)

In a geometric series \( \sum_{k=0}^{\infty} ar^k \), the form resembles \( \sum_{k=1}^{\infty} \frac{2}{3^{k-1}} = 2 \sum_{k=0}^{\infty} \left(\frac{1}{3}\right)^k \). Check the common ratio: \( r = \frac{1}{3} \), which is less than 1, signaling convergence.

Step 3: Find the Sum of the Geometric Series (a)

The formula for the sum of a geometric series \( \sum_{k=0}^{\infty} ar^k \) is \( S = \frac{a}{1-r} \). Here, \( a = 2 \) and \( r = \frac{1}{3} \), so: \( S = \frac{2}{1-\frac{1}{3}} = \frac{2}{\frac{2}{3}} = 3 \).

Step 4: Separate and Simplify the Series (b)

Express the series as \( \sum_{k=1}^{\infty} \left( \frac{2^k}{6^k} + \frac{3^k}{6^k} \right) \), which is \( \sum_{k=1}^{\infty} \left(\frac{1}{3}\right)^k + \sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^k \). Both are geometric series.

Step 5: Calculate Each Series in b)

For \( \sum_{k=1}^{\infty} \left(\frac{1}{3}\right)^k \), the sum is \( \frac{\frac{1}{3}}{1-\frac{1}{3}} = \frac{1}{2} \). For \( \sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^k \), the sum is \( \frac{\frac{1}{2}}{1-\frac{1}{2}} = 1 \).

Step 6: Conclusion for Part (b)

Add both sums from part (b): \( \frac{1}{2} + 1 = \frac{3}{2} \). This indicates that the series converges and the total sum is \( \frac{3}{2} \).

Key Concepts

Understanding Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." This is why the series is termed geometric. For example, a series such as \( \sum_{k=0}^{\infty} ar^k \) is characterized by the first term \(a\) and the common ratio \(r\).
A geometric series can either converge or diverge, depending on the absolute value of the common ratio \(r\):

• If \(|r| < 1\), the series converges, meaning it approaches a finite sum as \(k\) approaches infinity.
• If \(|r| \geq 1\), the series diverges, implying it grows indefinitely or oscillates.

The concept of geometric series is pivotal in understanding complex series and sequences in mathematics.

Calculating the Sum of a Series

To find the sum of an infinite geometric series, mathematicians use the formula \( S = \frac{a}{1-r} \), where \(S\) represents the sum, \(a\) represents the first term, and \(r\) represents the common ratio.
This formula is derived from the nature of a geometric series and is only valid when the series converges (i.e., when \(|r| < 1\)).

Calculating the sum requires identifying the first term and the common ratio. For instance, in the series \( \sum_{k=0}^{\infty} ar^k \), if \(a = 2\) and \(r = \frac{1}{3}\), the sum is:
\[S = \frac{2}{1-\frac{1}{3}} = \frac{2}{\frac{2}{3}} = 3.\]
This calculation tells us that the infinite series will sum up to 3, showing how vital the formula is for practical computations.

Convergence Criteria for Series

Convergence of a series determines whether the infinite sum of its terms results in a finite number. The key factor in convergence for a geometric series is the common ratio \(r\).
To decide if a given series converges, follow these guidelines:

• If \(|r| < 1\), the series converges, indicating that the series approaches a finite limit.
• If \(|r| \geq 1\), the series diverges, meaning no finite sum can be ascertained.

These criteria simplify the process of analyzing series, as they provide a clear indicator of behavior based on \(r\). Understanding convergence is crucial for the application of series in real-world problems, ensuring that calculations and predictions remain grounded and accurate.

Exploring Infinite Series

Infinite series extend indefinitely and are a fascinating area of study in mathematics. A series is a sum of terms, and an infinite series goes on without end.
Important concepts regarding infinite series include:

• A convergent infinite series has a definite finite sum, determined by the series' structure and characteristics, like the common ratio in geometric series.
• A divergent infinite series does not settle down to a specific value, often growing infinitely large or failing to approach a stable sum.

Infinite series are prevalent in various fields of science and engineering, where they model processes or phenomena that naturally extend beyond finitely calculable boundaries, such as in physics or signal processing. Mastery of infinite series concepts broadens one's ability to understand and apply higher-level mathematical concepts effectively.