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Magazine Chapter 3: Problem 1
Start from \(x_{0}=6\) and \(x_{0}=2\). Compute \(x_{1}, x_{2}, \ldots\) to test convergence: $$ x_{n+1}=x_{n}^{2}-\frac{1}{2} $$
Short Answer
Expert verified
The sequences starting from both initial points diverge.
Step by step solution
01
Compute x1 from x0 = 6
To find \(x_1\) using the formula \(x_{n+1} = x_n^2 - \frac{1}{2}\), substitute \(x_0 = 6\):\[ x_1 = 6^2 - \frac{1}{2} = 36 - \frac{1}{2} = 35.5 \]
02
Compute x2 from x1 = 35.5
Substitute \(x_1 = 35.5\) into the formula to compute \(x_2\):\[ x_2 = 35.5^2 - \frac{1}{2} \]\( 35.5^2 = 1260.25 \), so \[ x_2 = 1260.25 - \frac{1}{2} = 1259.75 \]
03
Compute x3 from x2 = 1259.75
Substitute \(x_2 = 1259.75\) into the formula to compute \(x_3\):\[ x_3 = 1259.75^2 - \frac{1}{2} \] \( 1259.75^2 = 1586989.0625 \), so \[ x_3 = 1586989.0625 - \frac{1}{2} = 1586988.5625 \]
04
Compute x1 from x0 = 2
Now evaluate a new sequence starting from \(x_0 = 2\). Use the same formula:\[ x_1 = 2^2 - \frac{1}{2} = 4 - \frac{1}{2} = 3.5 \]
05
Compute x2 from x1 = 3.5
Substitute \(x_1 = 3.5\) into the formula to find \(x_2\):\[ x_2 = 3.5^2 - \frac{1}{2} \]\( 3.5^2 = 12.25 \), so \[ x_2 = 12.25 - \frac{1}{2} = 11.75 \]
06
Compute x3 from x2 = 11.75
Substitute \(x_2 = 11.75\) into the formula to find \(x_3\):\[ x_3 = 11.75^2 - \frac{1}{2} \]\( 11.75^2 = 138.0625 \), so \[ x_3 = 138.0625 - \frac{1}{2} = 137.5625 \]
07
Observation on Convergence
The sequences from both starting points show increasingly larger terms and do not seem to be approaching a fixed value. Thus, the sequences diverge instead of converging.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergent and divergent sequences
In mathematics, sequences play a crucial role in understanding the behavior of functions and series. A sequence is a set of numbers arranged in a specific order. They are pivotal in calculus and analysis.
- Convergent Sequences: A sequence is said to be convergent if its terms approach a specific number as you move towards infinity. Simply put, beyond a certain point, all terms get arbitrarily close to a particular value.
- Divergent Sequences: Conversely, a sequence is divergent if its terms do not settle down to a specific number. They might grow without bounds, bounce, or oscillate.
Iterative sequences
Iterative sequences are constructed by applying a particular formula or recurrence relation repeatedly. Their purpose is to observe how the sequence evolves through iteration.
- Definition: An iterative sequence is defined such that each term is a transformation of the previous one using a consistent formula, like \(x_{n+1} = x_n^2 - \frac{1}{2}\).
- Usage: They are widely used in numerical methods to solve equations, approximate functions, and understand complex dynamics.
Convergence testing
Convergence testing is a mathematical process to determine if a sequence or series converges. It's crucial because convergence implies that a series or sequence settles to a point, providing meaningful limits or solutions.
- Importance: Knowing if a sequence converges helps in understanding the long-term behavior and stability of a system or function.
- Techniques: Common methods include limit testing, applying specific convergence tests like the Ratio Test, and using bounding properties.
