Staircase diagrams: Inward ladder

Occurs when 0 < f’(w) < 1

Where w is the root and f(x) is the convergent function

This will successfully obtain the root

Staircase diagrams: Outward ladder

Occurs when f’(w) > 1

Where w is the root and f(x) is a divergent function

This won’t obtain the root

Cobweb diagrams: Inward spiral

Occurs when -1 < f’(w) < 0

Where w is the root and f(x) is a convergent function

This will successfully obtain the root

Cobweb diagrams: Outward spiral

Occurs when f’(w) < -1

Where w is the root and f(x) is the divergent sequence

Won’t obtain the root

Drawing cobweb/staircase diagrams

Start with a point and draw a line to the curve, then across to the line then to the curve etc

Always start with the curve!

Using the derivative of the iterative function f(x) to determine if the process will converge or diverge

In order for the sequence to converge then:

-1 < f’(w) < 1

Where f’(w) is the derivative of the iterative function at w

Where w is the root

Alternating sequence

Oscillating sequence

An oscillating sequence which oscillates around 0 - so the terms are alternately positive and negative

A sequence that is alternately greater then small than a given value

Periodic sequence

A sequence that consists of a repeating pattern of numbers

Convergent sequence

Divergent sequence

A sequence which approaches a definite value

A sequence which doesn’t approach a definite value

Condition for a geometric progression to be convergent

the common ratio, r :

-1 < r < 1

Exponential growth and decay

When the rate of growth/decay is directly proportional to the quantity present

How can you tell which iterative sequence will converge more rapidly?

Magnitude of the gradient is closest to the root at the given starting point