Problem 12 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Sinn dieser letzten Aufgabe ist es, Sie selbständig entscheiden zu lassen, welches der zahlreichen Kriterien für Konvergenz Sie auf eine gegebene Reihe anwenden müssen. a) Für welche a \(\in \mathbb{R}\) konvergiert die Reihe \(\sum_{\mathrm{k}=1}^{\infty} \frac{1}{(a+k)(a+k+1)}\) was ist ihre Summe? b) Folgern Sie aus der Konvergenz von \(\sum_{\mathrm{k}=1}^{\infty} \frac{1}{k(k+1)}, d \mathrm{a} \beta\) die Reihe \(\sum_{\mathrm{k}=1}^{\infty} \frac{1}{k^{2}}\) konvergiert. C) Untersuchen sie die Reihen \(\sum_{k=1}^{\infty} \frac{1}{k^{a}}, a \in \mathbb{Z}\), auf Konvergenz und Divergenz.

Step-by-Step Solution

Answer

a) The series converges for \(a > -1\) with sum \(\frac{1}{a+1}\); b) It shows \(\sum \frac{1}{k^2}\) converges; c) \(\sum \frac{1}{k^a}\) converges for \(a > 1\).

Step 1: Understand the General Term and Rearrange

The given series is \[ \sum_{k=1}^{\infty} \frac{1}{(a+k)(a+k+1)}. \]For part (a), let's rewrite the general term using partial fraction decomposition.\[\frac{1}{(a+k)(a+k+1)} = \frac{1}{a+k} - \frac{1}{a+k+1}.\]This is a telescoping series.

Step 2: Determine Conditions for Convergence (Part a)

The convergence of the series depends on whether the partial sums converge as more terms are added.Since it's a telescoping series, many terms will cancel out and we're left with:\[\sum_{k=1}^{n} \left( \frac{1}{a+k} - \frac{1}{a+k+1} \right) = \frac{1}{a+1} - \frac{1}{a+n+1}.\]This converges as \(n \to \infty\) if \(a > -1\). The sum then converges to \(\frac{1}{a+1}\).

Step 3: Investigate the Implication for Other Series (Part b)

The series \[ \sum_{k=1}^{\infty} \frac{1}{k(k+1)} \]can be written as a telescoping series:\[\frac{1}{k} - \frac{1}{k+1}.\]This converges, and thus implies the convergence of \[ \sum_{k=1}^{\infty} \frac{1}{k^2} \]since if a series similar to a known convergent series can be shown to have terms that are smaller in magnitude, it converges.

Step 4: Investigate Series Convergence (Part c)

The series\[ \sum_{k=1}^{\infty} \frac{1}{k^a} \]is known as a p-series.- It converges if \(a > 1\) (since the exponent \(p = a > 1\) in a p-series ensures convergence).- It diverges if \(a \leq 1\) due to the p-series test, which states series diverge for \(p \leq 1\).

Key Concepts

Reihenuntersuchung

When analyzing series for convergence, we use a method called 'Reihenuntersuchung,' or series investigation. This involves applying various criteria to determine if a series converges or diverges. A vital part of this process is understanding the general term of the series and exploring its properties.

• Rewriting the series: Sometimes, breaking down the series into simpler parts using identities or transformations makes it easier to apply convergence tests.
• Testing for convergence: Common tests include the comparison test, ratio test, and root test, each offering insights into whether a series will tend to a finite sum or not.

Proper Reihenuntersuchung allows one to select the appropriate convergence criterion, depending on the nature of the series at hand, such as whether the series is geometric, arithmetic, or involves complex functions.
With systematic methodology, one can predict whether piling up infinite terms results in a finite total.

Partialbruchzerlegung

Partial fraction decomposition, or "Partialbruchzerlegung," is a powerful algebraic tool used to simplify complex rational expressions. It is particularly useful in series analysis, where it helps break down terms into sums of simpler, easily recognizable fractions.
This process involves expressing a fraction \[\frac{1}{(a+k)(a+k+1)}\] as the sum of two fractions, like \[\frac{1}{a+k} - \frac{1}{a+k+1}.\]

• Each term in the decomposition cancels out parts of the series, facilitating the identification of patterns such as telescoping.
• Utilizing partial fraction decomposition simplifies the process of analyzing convergence, as simpler fractions can be more readily evaluated.

By transforming a series into a more manageable form, Partialbruchzerlegung often uncovers intrinsic properties of the series, rendering it more approachable for convergence tests.

p-Reihe

The p-series, or "p-Reihe," is a fundamental concept in understanding series convergence. It is characterized by its general form:\[\sum_{k=1}^{\infty} \frac{1}{k^a}\]where "a" represents a real number parameter that dictates the series' behavior.

• If \(a > 1\), the p-Reihe converges. This is because the terms shrink sufficiently fast as \(k\) increases, making the total finite.
• If \(a \leq 1\), the series will diverge. In these cases, the terms do not diminish quickly enough, causing the sum to swell indefinitely.

Recognizing a p-series is crucial, as it provides a quick way to apply convergence tests by simply evaluating the exponent. This form of series serves as a baseline for understanding more complex series' convergent or divergent behavior.

Teleskopsumme

A telescoping series, or "Teleskopsumme," is a series where many terms cancel each other out, pushing the focus to only a few terms. This makes it easier to determine convergence and find the sum. For example, consider the series: \[\sum_{k=1}^{n} \left( \frac{1}{a+k} - \frac{1}{a+k+1} \right).\]

• When expanded, intermediate terms like \( \frac{1}{a+k} \) and \( -\frac{1}{a+k} \) cancel out, simplifying the sum.
• The result is typically a difference of a few remaining terms, which makes calculating limits for large \(n\) very manageable.

Since only a finite number of terms significantly affect the result, determining the convergence of a Teleskopsumme becomes a straightforward task. This property is especially advantageous in problems where direct summation would be cumbersome or impossible.