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Sequences are complicated.

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[MUSIC]

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It's not so easy to find the limit of a
sequence.

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The logistic map provides one source of
rich, and by rich I mean very complicated.

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Examples.
Let's see how these are defined.

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Here's a sequence I want to consider.

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It's actually a family of sequences, which
depend on some parameter r.

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Let's call r, 2.5 for the time being.
And I'll pick an initial

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value for my sequence.
Let's make it 0.25 for now.

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And then here's the recursive definition
of the sequence.

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A sub n plus one will be this constant r
times the

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previous term a sub n times one minus a
sub n.

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And by varying r and by varying the
initial value

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I'll get a whole bunch of different
sequences to consider.

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Let's try it with those values.
Let's try it with r equals 2.5.

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So here's an example.

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I start with the initial value, 0.25 and
each subsequent value

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is 2.5 times the previous value times one
minus the previous value.

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We have graphed these values, right?

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The first is at 0.25, then it goes up

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a bit, then it starts sort of levelling
off here.

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And indeed, if we look at this
numerically, and

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if the first value is exactly 0.25, and
then as I run down through the

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sequence, it seems like the values are
getting closer and closer to about 0.6.

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And in this case the limit of the sequence
is in fact 0.6.

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What if we start with a different initial
value?

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What if instead of a sub one equals 0.25,
we make a sub one equal something else.

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So the recursive formula we're using is a
sub n plus one is

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2.5 times a sub n times one minus a sub n.
And

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before, we were starting with a sub one
equal to 0.25.

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Let's start with 0.8 and see what happens
in that case.

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So if I start with 0.8, I could rewrite
0.8 as 4

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5ths, and instead of writing 2.5 I'll
write five

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halves times a n times one minus a n.

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Just like working with the fractions
instead of the decimals.

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So let's use this to calculate the next
term.

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What a sub two?

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Well, according to this formula, it's five
halves times

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a sub one, times one minus a sub one.

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But I've got a sub one, it's 4 5ths.

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So that's five halves times 4 5ths minus
one minus 4 5ths, which is 1 5th.

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Now, 1 5th and this five can cancel.
I can get rid of these.

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And this four and this two, become a two
in the numerator, so I've just got 2 5ths.

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So a sub two is 2 5ths.
Now I can calculate a sub three.

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[SOUND]

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So a sub three will be five halves again,
times a sub

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two, times one minus a sub two.
And, in this case.

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What is, a sub two, we just calculated,
it's 2 5ths.

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So this is five halves times 2 5ths times
one minus 2

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5ths, which is 3 5ths.
But now I've got five halves and

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2 5ths, and those cancel, so a sub three
is just 3

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5ths.
Now, let's calculate a sub four.

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Well, a sub four is, again, five halves
times

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a sub three times one minus a sub three.

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That's five halves times a sub three is 3
5ths.

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One minus 3 5ths is 2 5ths.

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But now I've

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got 5 halves and 2 5ths.
Those cancel.

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So a sub four is just 3 5ths.
Or what's a sub five?

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Well, a sub 5 is also 3 5ths.

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I'm going to use the exact same formula
here but I'm again just

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going to plug in 3 5ths and I'm going to
get 3 5ths out.

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So a sub five is 3 5ths a sub six is 3
5ths, right.

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The point is that this sequence is just
constant from

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here on out.
And that means that, in this case the

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limit as n approaches infinity of the
sequence is just 3 5ths,

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just like in the case when I started with
0.25, alright.

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It happened in this case that even when I
start with 0.8,

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when I use this formula I again ended up
with the same limit.

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It's kind of interesting.
Let's try

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r equals 7 3rds.

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I'll again, have my sequence start with
0.25, but then each new term

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is 7 3rds times the old term, times one
minus the old term.

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And if we look at this graph, it looks
like the

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sequence is converging, and if we look at
the numbers, right.

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The first term in this sequence is 0.25
and it's

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about 0.4 and about 0.57 and it keeps on
going and

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it does indeed look like, the sequence is
getting closer and closer to something.

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This number might not be too meaningful to
you, but you can

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in fact show that the limit of this
sequences is 4 7ths.

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And indeed, this number is about 4 7ths.

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So changing the value of r seems to affect
the limit.

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Let's try r equals 3.25.

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So again my sequence will start

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with 0.25, but now the r value is 3.25.
What does the sequence look like?

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Well, I can graph a bunch of terms of the
sequence, and it starts with

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a 0.25 and it goes up, and then it seems
to bounce between two values.

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And indeed, the numerical evidence
supports that same conclusion.

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Here's the first value, 0.25 and it's
about 0.6,

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0.7, 0.5, 0.8, 0.5, 0.8, 0.5.

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And since it's been flip-flopping between
these two values.

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And since it's flip-flopping between those
two values, the limit doesn't exist.

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Let's try another example.

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Let's try r equals 3.7.

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The same initial value 0.25, but now 3.7
is my value for r.

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What does this sequence look like?
Well,

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I can start graphing the terms in this
sequence and wow.

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I mean, it just looks like garbage.

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There just doesn't seem to be any pattern
at all, you know?

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And there's maybe moments when it looks
like things

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are getting better, but then it suddenly
breaks apart again.

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And if you look at the numbers, right.

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Numerically things don't look so great
here either.

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I mean 0.25, that's the initial value but
as you look through

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these numbers, it doesn't look like any
sort of pattern.

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These really coming out.

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What is going on here?

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Well changing the value of r doesn't

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just change quantitative features of the
sequence.

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It changes qualitative features of the
sequence.

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Depending on the value of r, this sequence
might converge, might have a limit.

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It might flip-flop between a couple

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values, might flip-flop between four
different

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values, it might just move all over the
place

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and not really have any kind of
discernible pattern.

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And it all boils down to this value of r
in a seemingly mysterious way.

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But that's not to say that the sequence

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can't be understood, that, that it can't
be studied.

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Alright.

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Maths isn't just random, right, I mean
there's structure to this thing,

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and with more work you can really start
digging in, to the very

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complicated structure that appears, even
something as

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seemingly simple as just the sequence of
numbers.

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[SOUND]