1 and yn+l = V2 + Yn for n > 1. Show (ym) is monotone, and Let yn bounded, and compute its limit.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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### Question 5

Let \( y_1 = 1 \) and \( y_{n+1} = \sqrt{2 + y_n} \) for \( n \geq 1 \). Show \((y_n)\) is monotone, and bounded, and compute its limit.

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In this question, we are given a sequence defined recursively by \( y_1 = 1 \) and \( y_{n+1} = \sqrt{2 + y_n} \) for \( n \geq 1 \). We are tasked with proving three properties of this sequence: that it is monotone, that it is bounded, and finding its limit.

1. **Monotonicity**:
   Determine whether the sequence is increasing or decreasing.

2. **Boundedness**:
   Prove that there is an upper bound to this sequence.

3. **Limit**:
   Calculate the limit of the sequence as \( n \) approaches infinity.

This problem involves using techniques from mathematical analysis, such as induction, to demonstrate the properties of the sequence.

- **Monotonicity**: By definition, a sequence is monotone if it is either entirely non-increasing or non-decreasing.
- **Boundedness**: A sequence is bounded if there exists some number M such that every term in the sequence is less than or equal to M.
- **Limit Calculation**: The limit of the sequence can be found by solving the resulting equation when setting \( y = y_{n+1} = y_n \).
Transcribed Image Text:### Question 5 Let \( y_1 = 1 \) and \( y_{n+1} = \sqrt{2 + y_n} \) for \( n \geq 1 \). Show \((y_n)\) is monotone, and bounded, and compute its limit. --- In this question, we are given a sequence defined recursively by \( y_1 = 1 \) and \( y_{n+1} = \sqrt{2 + y_n} \) for \( n \geq 1 \). We are tasked with proving three properties of this sequence: that it is monotone, that it is bounded, and finding its limit. 1. **Monotonicity**: Determine whether the sequence is increasing or decreasing. 2. **Boundedness**: Prove that there is an upper bound to this sequence. 3. **Limit**: Calculate the limit of the sequence as \( n \) approaches infinity. This problem involves using techniques from mathematical analysis, such as induction, to demonstrate the properties of the sequence. - **Monotonicity**: By definition, a sequence is monotone if it is either entirely non-increasing or non-decreasing. - **Boundedness**: A sequence is bounded if there exists some number M such that every term in the sequence is less than or equal to M. - **Limit Calculation**: The limit of the sequence can be found by solving the resulting equation when setting \( y = y_{n+1} = y_n \).
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