Problem 5 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

a) Finden Sie neben der harmonischen Reihe ein Beispiel einer divergenten Reihe \(\sum_{k=1}^{\infty} a_{k}\), für die \(\lim _{n \rightarrow \infty} a_{n}\) existiert. b) Zeigen Sie mit Hilfe von Satz (8.10) nochmals, daß die geometrische Reihe \(\sum^{\infty} x^{k-1}\) für \(|x|>1\) divergiert. \(k=1\) C) Konvergiert oder divergiert die Reihe $$ \sum_{k=1}^{\infty}\left(\frac{1}{k}+\frac{1}{3^{k}}\right) ? $$

Step-by-Step Solution

Answer

a) An example is \( \sum (-1)^k \). b) The series diverges for \(|x|>1\). c) The series diverges.

Step 1: Identify Divergent Series

Consider the harmonic series as an example: \[ \sum_{k=1}^{\infty} \frac{1}{k} \]It is a known divergent series and the limit of its terms tends to zero, \( \lim_{n \to \infty} \frac{1}{n} = 0 \).Another example is the series:\[ \sum_{k=1}^{\infty} c_k \] where \( c_k = (-1)^k \). The limit \( \lim_{n \to \infty} c_n = 0 \), but it does not converge absolutely, thus it diverges.

Step 2: Geometric Series and Divergence

The geometric series \( \sum^{\infty} x^{k-1} \) with \( |x| > 1 \) is divergent. According to the formula for the sum of a geometric series \( S = \frac{a}{1 - r} \), it only converges for \( |r| < 1 \). Since \(|x| > 1\), the common ratio \(r = x\), and as \(k\) increases, the terms of the series grow, thus the series diverges.

Step 3: Analyze the Given Series for Convergence

Consider the series \( \sum_{k=1}^{\infty} \left( \frac{1}{k} + \frac{1}{3^k} \right) \). This can be split into two separate series, \( \sum_{k=1}^{\infty} \frac{1}{k} \) which is a harmonic series known to diverge, and \( \sum_{k=1}^{\infty} \frac{1}{3^k} \) which is a convergent geometric series since \( |\frac{1}{3}| < 1 \). However, the divergence of the harmonic series dominates, thus the overall series diverges.

Key Concepts

Harmonic Series

The harmonic series is one of the most famous examples of a divergent series. It is defined as:\[ \sum_{k=1}^{\infty} \frac{1}{k} \]Each term of the series adds a fraction of decreasing size, but that doesn't lead to the sum approaching a finite value. Instead, as you add more terms, the sum grows indefinitely. Despite each fraction getting smaller, they never decrease rapidly enough for the sum to settle.To understand why this series is divergent, try considering a simplistic grouping of terms. For instance:

• Sum the first term by itself.
• Sum the next two terms \( \frac{1}{2} + \frac{1}{3} \) which exceeds \( \frac{1}{2} \).
• The next four terms \( \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} \) also exceed \( \frac{1}{2} \).
• This pattern shows that the sum keeps growing as you add broad chunks of terms, each exceeding a fixed value.

Consequently, the harmonic series diverges because these growing chunks do not taper off to zero effectively.

Geometric Series

A geometric series is recognized for its definite pattern where each term is a constant multiple of the previous one. The general form is:\[ \sum_{k=1}^{\infty} ar^{k-1} \]where \(a\) is the initial term, and \(r\) is the common ratio. This series converges if the absolute value of the common ratio \(|r|\) is less than 1.Why does it need to be \(|r| < 1\)? Here's why:When \(|r| < 1\), each subsequent term becomes smaller, approaching zero. The series can be summed using the formula:\[ S = \frac{a}{1 - r} \]This formula can't be applied when \(|r| \geq 1\), as the terms don't decrease towards zero. The series diverges in such cases, meaning the sum approaches infinity.An example to consider is when \(|x| > 1\), the geometric series \( \sum^{\infty} x^{k-1} \) diverges. Because \(r = x\), and since \(|x| > 1\), the whole series cannot settle down to a finite sum.

Series Convergence and Divergence

To determine whether a series converges or diverges, one must analyze the behavior of the series as more terms are added. A series converges if the sum of its infinite terms approaches a specific finite number, while it diverges if the sum becomes infinite.There are several strategies to analyze convergence or divergence:

• Comparison Test: Compare it with a known convergent or divergent series. If a series has terms bigger than a known divergent series, it too diverges. Conversely, if its terms are smaller than a convergent series, it may converge.


• Geometric Series Test: Look at the common ratio \(r\). If \(|r| < 1\), the series converges, otherwise it diverges.


• Harmonic Series Test: The harmonic series is a key example of a divergent series. If a series resembles the harmonic series, where terms diminish but do not diminish fast enough, it may be divergent.

In practical terms, you can often split complex series into parts and analyze each part. For instance, with mixed series like \( \sum_{k=1}^{\infty} \left( \frac{1}{k} + \frac{1}{3^k} \right) \), you break it into a known divergent part (harmonic) and convergent part (geometric), determining overall behavior from the dominant divergent component.