Math Problem Statement

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Solution

The image contains a mathematical exercise related to sequences and convergence. Let me transcribe and analyze the problem to provide a detailed solution.

Transcription of the Problem:

Exercise 1:
Sequence unu_n:

u_0 = -1 \\ u_{n+1} = \frac{1}{1 - u_n}, \, \text{for } n \in \mathbb{N}. \end{cases}$$ 1. **Study of the sequence $$u_n$$:** a) Show by induction that $$u_n \neq 0 \, \forall n$$. b) Prove that the sequence $$(u_n)$$ is strictly negative ($$u_n < 0 \, \forall n$$). c) Prove that $$(u_n)$$ is strictly increasing ($$u_{n+1} > u_n \, \forall n$$). 2. **Convergence of $$u_n$$:** a) Express $$u_{n+1}$$ in terms of $$u_n$$ and study its behavior as $$n \to \infty$$. b) Determine whether $$(u_n)$$ converges, and if so, find its limit $$L$$. 3. **Definition of sequence $$v_n$$:** Define $$v_n = u_n + 1$$. a) Express $$v_{n+1}$$ in terms of $$v_n$$. b) Prove that $$v_n$$ converges to a limit. 4. **Graphical Representation:** a) Graph the sequence $$(u_n)$$. b) Interpret the evolution of $$(u_n)$$ based on its properties. --- **Solution Approach:** Would you like me to proceed step by step for each sub-question, starting with 1a? Let me know if you'd prefer a specific focus.

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Math Problem Analysis

Mathematical Concepts

Sequences
Convergence
Monotonicity
Induction

Formulas

Recursive sequence formula: u_{n+1} = 1 / (1 - u_n)
v_n = u_n + 1

Theorems

Principle of Mathematical Induction
Convergence criteria for monotonic sequences

Suitable Grade Level

Undergraduate or Advanced High School (Grade 11-12)

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