Learn7357Paper 2 · Pure and MechanicsPure content (sections A–J)Rearrangements, staircases and cobwebs

Pure content (sections A–J) · Subtopic 58 of 66

Rearrangements, Staircases and Cobwebs

One equation can be rearranged into $x=F(x)$ in many ways, and they do not all work. See why some iterations close in on a root, some zigzag towards it, and some run away, and learn to find the equation a limit satisfies.

Cambridge 9709 Paper 2Cambridge 9709 Paper 3Edexcel YMA01 WMA13Edexcel 9MA0 Paper 1Edexcel 9MA0 Paper 2AQA 7357 Paper 1≈ 40 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Rearrange an equation $f(x)=0$ into the form $x=F(x)$ in more than one way.
Find the equation, and sometimes the exact value, that the limit of an iteration satisfies.
Draw and read staircase and cobweb diagrams.
Use the gradient $F'(\alpha)$ to explain why an iteration converges or diverges.

Pause and think

$x^3-2x-5=0$ can be written as $x=\sqrt[3]{2x+5}$ or as $x=\dfrac{x^3-5}{2}$ . Both are true at the root. Start each iteration at $x_1=2$ . What happens?

Section 2 of 14

Core idea

Start hereWhy some iterations run away

$x^2-3x+2=0$ has roots $1$ and $2$ . The iteration $x_{n+1}=\frac{x_n^2+2}3$ finds $1$ beautifully, but start it near $2$ and it flees: $2.2,2.28,2.40,2.59\ldots$ The deciding factor is the gradient of $g$ at the root.

The big idea

$x_{n+1}=g(x_n)$ converges to $\alpha$ (from nearby) if $|g'(\alpha)|<1$ and diverges if $|g'(\alpha)|>1$ . $0<g'<1$ : staircase; $-1<g'<0$ : cobweb.

1. The error argument

How errors change each step

$e_n=x_n-\alpha$
Error at step $n$ .
$e_{n+1}=g(x_n)-g(\alpha)$
$x_{n+1}=g(x_n)$ and $\alpha=g(\alpha)$ .
$\approx g'(\alpha)(x_n-\alpha)=g'(\alpha)e_n$
Near $\alpha$ the curve is almost its tangent.

Each error is multiplied by about

g'(\alpha)

. If

|g'(\alpha)|<1

errors shrink; if

>1

they grow.

Cobweb for x(n+1) = cos x(n), starting at 0

Estimates

0,\ 1,\ 0.540,\ 0.858,\ 0.654,\ 0.794,\ 0.701\ldots

land on alternate sides of the root and spiral in.

Starting at 2.2: the staircase runs away from the root 2

Near

x=2

the curve is steeper than

y=x

, so each step overshoots by more:

2.2,\ 2.28,\ 2.40,\ 2.59,\ 2.90\ldots

Worked exampleWhich root can $x_{n+1}=\frac{x_n^2+2}3$ find?

Decide for each root of $x^2-3x+2=0$ .

Goal: Converge or diverge.
Strategy: Compute $g'$ at each root.

$g'(x)=\tfrac{2x}3$
Differentiate.
$g'(1)=\tfrac23<1$
Converges (staircase).
$g'(2)=\tfrac43>1$
Diverges.

AnswerFinds

1

; can never find

2

✓ CheckMatches the pictures ✓.

Worked example $x_{n+1}=\cos x_n$

Describe the convergence.

Goal: Type.
Strategy: Sign and size of $g'$ .

$1,\ 0.5403,\ 0.8576,\ 0.6543,\ 0.7935,\ 0.7014,\ldots$
Iterates jump either side.
$g'(0.739)=-\sin0.739=-0.67$
Between $-1$ and $0$ .

AnswerCobweb convergence to

0.739

✓ CheckErrors alternate sign and shrink by about

0.67

each step ✓.

Pause and think

Will $x_{n+1}=x_n^3-5$ find the root near 1.9 of $x^3-x-5=0$ ?

Summary

$|g'(\alpha)|<1$ converges.
Sign of $g'$ : staircase or cobweb.

Converge: Approach a limit.
Cobweb: Spiral picture of oscillating convergence.

HistoryCobwebs and pig prices. Cobweb diagrams come from economics. In the 1930s economists noticed that pig prices rose and fell in a regular cycle: high prices one year led farmers to breed more pigs, which pushed the price down the next year, and so on. Nicholas Kaldor named it the cobweb theorem in 1934, after the shape of the diagram.

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