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This document contains homework exercises on sequences ("Folgen") in German. Let's break down the tasks to understand what each question requires:

### Problem 1: Recursive Sequences ("Rekursive Folgen")
We are given the recursive sequence \( a_1 = 1 \) and \( a_{n+1} = 1 + \frac{a_n^2}{4} \). The problem has three parts:

1. **(a)** Show by mathematical induction that \( 1 \leq a_n \leq 2 \) for all \( n \in \mathbb{N} \).
   
2. **(b)** Show that the sequence \( (a_n)_{n \in \mathbb{N}} \) is monotonically increasing.
   
3. **(c)** Conclude that \( (a_n)_{n \in \mathbb{N}} \) converges and determine its limit.

### Problem 2: Convergence of Sequences ("Konvergenz von Folgen")
The second problem contains three different sequence convergence tasks:

1. **(a)** Let \( a > 0 \). Using the Squeeze Theorem ("Squeeze-Theorem"), show that \( \lim_{n \to \infty} \sqrt{1 + \frac{a}{n}} = 1 \).
   
2. **(b)** Let \( a > 0 \). Determine the limit of the sequence \( \left( \sqrt{n^2 + a n - n} \right)_{n \in \mathbb{N}} \).

3. **(c)** We consider the sequence \( (a_n)_{n \in \mathbb{N}} \) defined by \( a_n = \frac{1}{1+n} + \frac{1}{2+n} + \dots + \frac{1}{n+n} \). Show that \( (a_n) \) is monotonically decreasing and bounded, and therefore convergent. Determining the exact limit is not required, but giving a tip on finding it would be helpful.

Would you like a solution or detailed explanation for a specific part of this exercise set? 

### Related Questions
1. How do we apply mathematical induction to prove inequalities in recursive sequences?
2. What are the steps to show that a sequence is monotonic and bounded?
3. How can we apply the Squeeze Theorem in limit proofs for sequences?
4. What methods help in finding limits of sequences involving square roots?
5. How do we establish convergence without finding an explicit limit?

### Tip
For recursive sequences, proving that the sequence is bounded and monotonic often helps in establishing convergence.

Hausübungen

1. Rekursive Folgen
Wir betrachten die rekursive Folge a_1 = 1, a_{n+1} = 1 + \frac{a_n^2}{4}.

(a) Zeigen Sie durch vollständige Induktion, dass 1 \leq a_n \leq 2 für alle n \in \mathbb{N} gilt.
(b) Zeigen Sie damit, dass die Folge (a_n)_{n \in \mathbb{N}} monoton wachsend ist.
(c) Folgern Sie, dass (a_n)_{n \in \mathbb{N}} konvergiert und bestimmen Sie den Grenzwert.

2. Konvergenz von Folgen

(a) Sei a > 0. Zeigen Sie mithilfe des Squeeze-Theorems, dass \lim_{n \to \infty} \sqrt{1 + \frac{a}{n}} = 1 gilt.
(b) Sei a > 0. Bestimmen Sie den Grenzwert der Folge \left( \sqrt{n^2 + an - n} \right)_{n \in \mathbb{N}}.
(c) Wir betrachten die Folge (a_n)_{n \in \mathbb{N}} mit a_n = \frac{1}{1+n} + \frac{1}{2+n} + \dots + \frac{1}{n+n}. Zeigen Sie, dass (a_n) monoton wachsend und beschränkt, also konvergiert ist.

Explore exercises on recursive sequences and convergence in calculus. This includes proving inequalities by induction, showing monotonicity and boundedness in sequences, applying the Squeeze Theorem, and finding sequence limits. Suitable for undergraduate math students.

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