A recursive definition of π/ 2 If you start with x1=1 and define the subsequent terms of {xn} by

step 1 Hello, welcome to this lesson. In this lesson we've been given xN, that is the generating sequen

A recursive definition of π/ 2 If you start with x1=1 and...

A recursive definition of $\pi / 2$ If you start with $x_{1}=1$ and define the subsequent terms of $\left\{x_{n}\right\}$ by the rule $x_{n}=x_{n-1}+\cos x_{n-1},$ you generate a sequence that converges rapidly to $\pi / 2 .$ (a) Try it. (b) Use the accompanying figure to explain why the convergence is so rapid.

A recursive definition of $\pi / 2$ If you start with $x_{1}=1$ and
define the subsequent terms of $\left\{x_{n}\right\}$ by the rule
$x_{n}=x_{n-1}+\cos x_{n-1},$ you generate a sequence that converges
rapidly to $\pi / 2 .$ (a) Try it. (b) Use the accompanying figure to
explain why the convergence is so rapid.

Key Concepts

Fixed Point Iteration

Fixed point iteration is a method for finding a point that remains unchanged by a function. The sequence converges to a fixed point which, for this recursion, is the value that satisfies the equation obtained when the recursive definition is replaced by the fixed point condition. Here, the fixed point is related to the zero of the cosine function.

Properties of the Cosine Function

The cosine function plays a key role in the recursion, as its behavior determines the dynamics of the iterative process. Notably, cosine reaches zero at certain points, and in this instance, the argument for which cosine is zero defines the fixed point. Its properties, including its periodicity and the rate at which it approaches zero near these points, are central to understanding the rapid convergence observed.

Graphical Analysis of Convergence

Graphical analysis, such as investigating the tangent line to the iterative function's graph at the fixed point, provides insight into why the convergence is rapid. The figure suggests that because the cosine component becomes small near the fixed point, the steps taken by the sequence become progressively larger in reducing the error, leading to a fast approach to the limit.

Recursive Sequences

A recursive sequence is one in which each term is defined in relation to one or more previous terms. In this context, the sequence is defined by adding the cosine of the previous term to itself, making it a simple iterative process that can be used to approximate a value by repeated application of the same rule.

Stability and Rate of Convergence

The convergence of an iterative process is influenced by the derivative of the iterative function at the fixed point. In this case, the derivative vanishes at the fixed point, indicating that the usual linear term in error expansion is absent. This often results in superlinear or even quadratic convergence, meaning that the error decreases very rapidly as the sequence approaches the fixed point.

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