Today I want to illustrate why we want to study sequences,
and convergence, etc. I'll look at two Mathematica demos that
I've created in the last week or so. The first is an ancient
problem, really beautiful, and I created the demonstration
motivated by you and this course; the second arose completely
organically in the last few days, after a student pointed me to
an interesting Numberphile video on the web.
Sequences arise quite frequently, in ancient moments, and in
any moment. I hope that you'll enjoy them -- and maybe play
with them yourself!
One of the things to think about is the notion of
convergence, and the value of divergence!
- Archimedes approximates $\pi$
- Archimedes constructed a pair of sequences of
polygons to squeeze a circle, and obtained an
estimate of $2\pi$ by squeezing it between two
sequences of perimeters.
- He understood that the sequences had to converge
to the true value of $2\pi$.
- He just had to overcome his poor equipment for
continuing the experiment. He would have liked
Mathematica, rather than a stick and a lot of sand....
- The Golden Ratio
- A student pointed me to a an
episode of Numberphile on YouTube which finally
explained to me why we see Fibonacci numbers in flower,
pine cones, etc. It's wonderful, and if you watch it
first you'll see why I simply had to create this
Mathematica demo!
- We generate sequences of coordinates of "seeds",
$(x_k,y_k)$, which circle about the seed
producing center of a flower.
- We also generate a sequence of distances from the
seed-producing center, a sequence which
diverges, and pushes those coordinates to
diverge, as well. (Message: divergent
sequences can be useful, too!)
- It turns out that the "golden ratio" $\phi$
is the limit of a sequence, of the ratio of successive
Fibonacci numbers, which appear in the spiral arms of
the "flower" produced:
\[
\phi = \frac{1+\sqrt{5}}{2} =
\lim_{k\rightarrow\infty} \frac{F_{k+1}}{F_k}
\]
where the Fibonacci sequence is defined
recursively, by $F_1=1$, $F_2=1$, and
\[
F_{k} = F_{k-1}+F_{k-2}
\]
for $k=3,4,5,\ldots$.
- By the way, the limit above should remind you of
the ratio test, which we introduced this week!
- Long talking you through.