Geometric Series

Let a,r\in\mathbb{R} be constants. Let \{a_n\} be a sequence of real numbers with the following recursive definition:

a_n=\begin{cases}a & \text{ if }n=1\\ r\cdot a_{n-1}&\text{ if }n>1\end{cases}.

Therefore the sequence is given by:

a,ar,ar^2,ar^3,ar^4,\dots

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

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

S_n=a+ar+ar^2+ar^3+\cdots ar^{n-1}.

Here S_n is the sum of the first n terms.

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

r\cdot S_n=ar+ar^2+ar^3+\cdots ar^n

\Rightarrow a+r\cdot S_n-ar^n=S_n

\Rightarrow S_n(r-1)=a(r^n-1)

\displaystyle \Rightarrow S_n=\frac{a(r^n-1)}{r-1}.

Exercises

Assuming that |r|<1, find a formula for the geometric series

\displaystyle S_{\infty}=\lim_{n\rightarrow \infty}S_n.

Binary Numbers

Exercises

  • Write the following as fractions:

0.1_2,\,0.11_2,\,0.101_2.

  • Use infinite geometric series to show that:
    • 0.111\dots_2=1
    • 0.0111\dots_2=\frac12
    • 0.101010\dots_2=\frac23

Doubling Mapping

The doubling mapping D:[0,1)\rightarrow [0,1) is given by:

\displaystyle D(x)=\begin{cases}2x & \text{ if }x<1/2 \\ 2x-1 & \text{ of }x\geq 1/2\end{cases}.

Exercises

  • Find the first six iterates of the point x_0=\frac17 under D.
  • Find the first four iterates of the point

x_0=\frac{1}{2}+\frac{1}{2^2}+\frac{0}{2^3}+\frac{1}{2^4}=0.1101_2.

  • Where x has the binary representation

x = 0.a_1a_2a_3a_4a_5a_6a_7a_8\dots ,

write down expressions for D(x) and D^5(x).

  • Hence find points y, z \in [0, 1] such that y and z agree to 5 binary digits but D^N(y) and D^N(z) differ in the first binary digit for some N \in \mathbb{N}.
  • Describe the period-5 points of D.
  • Let w \in [0, 1] have a binary representation beginning w = 0.01001\dots  . Find a period-5 point \gamma of D such that w and \gamma agree to five binary digits.
  • Find a \delta \in [0, 1] such that there are iterates of \delta, D^{n_1}(\delta),D^{n_2}(\delta),D^{n_3}(\delta), with n_1, n_2, n_3 \in \mathbb{N}, that agree with 0.111 , 0.101, and 0.010, to three binary
    digits.

Sensitivity to Initial Conditions

Exercise

Let f(x)=4x\cdot (1-x). Where [0,1] is the set of states, and f:[0,1]\rightarrow [0,1] the iterator function, by looking at the first seven iterates of x_0=0.8 and y_0=0.81, show that this dynamical system displays sensitivity to initial conditions [HINT:4*ANS*(1-ANS)]

 

 

Advertisements