## Geometric Series

Let be constants. Let be a sequence of real numbers with the following recursive definition:

.

Therefore the sequence is given by:

Such a sequence is called a *geometric sequence with common ratio .*

When we add up the terms a sequence we have a *geometric sum:*

.

Here is the sum of the first terms.

We can find a formula for using the following ‘trick’:

.

*Exercises*

Assuming that , find a formula for the *geometric series*

.

## Binary Numbers

*Exercises*

- Write the following as fractions:

.

- Use infinite geometric series to show that:

## Doubling Mapping

The *doubling mapping* is given by:

.

*Exercises*

- Find the first six iterates of the point under .
- Find the first four iterates of the point

.

- Where has the binary representation

,

write down expressions for and .

- Hence find points such that and agree to 5 binary digits but and differ in the first binary digit for some .
- Describe the period-5 points of .
- Let have a binary representation beginning . Find a period-5 point of such that and agree to five binary digits.
- Find a such that there are iterates of , , with , that agree with 0.111 , 0.101, and 0.010, to three binary

digits.

## Sensitivity to Initial Conditions

*Exercise*

Let . Where is the set of states, and the iterator function, by looking at the first seven iterates of and , show that this dynamical system displays sensitivity to initial conditions [HINT:4*ANS*(1-ANS)]

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