Geometric progression, geometric series, and applications

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Geometric progression, geometric series, and
applications
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11/21/41/81/161/321/64 2
Is this, then, true or mere vain
fantasy? Euripedes
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REVIEW QUESTION If the series
converges, then

• an?8
• an?0
• an?l, where l is some finite number.
• there is no statement about the behavior of an.

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REVIEW QUESTION The numbers 0.3, 0.33, 0.333,
0.3333, 0.33333, are the partial sums of a
series that __________.

• True converges
• False converges
• True diverges
• False diverges

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We recognize the limit of the sequence of the
partial sums as the number 0.333333333333333333333
33333333333333333.. which is another way to
express the fraction 1/3 Formal proof Sum 3/10
3/1003/1000. (3/10)
(110-110-2) (3/10) 1 / (1-1/10)
1/3
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(No Transcript)
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John von Neumann
master of calculations
John von Neuman (1903-1957)
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A mathematician who knew a problem admitting one
quick but easily missed solution and one long but
straightforward solution was posing the problem
to various mathematicians to see how they would
respond. They all missed the quick solution and
tried to solve it via the long one. So, he
wanted to see von Neumann struggle with it too.
He posed the question, and von Neumann responded
with the right answer in a few seconds. "Interesti
ng," said the mathematician. "Most people try
to sum the infinite series." "What do you mean?"
von Neumann replied. "That's how I did it."
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THE PROBLEM
Two trains 200 miles apart are moving toward each
other each one is going at a speed of 50 miles
per hour. A fly starting on the front of one of
them flies back and forth between them at a rate
of 75 miles per hour. It does this until the
trains collide and crush the fly to death. What
is the total distance the fly has flown?
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CAN YOU SEE THE SOLUTION?

• The fly has flown
• 75 miles
• 100 miles
• 125 miles
• 200 miles
• none of the above

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The easy solution goes as follows Since the
trains are 200 miles apart and each train is
traveling 50 miles an hour, it takes 2 hours for
the trains to collide. Therefore the fly will
fly for two hours. Since the fly is flying at a
rate of 75 miles per hour, the fly will travel a
distance of 150 miles.
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There is a harder way to answer the question The
fly actually hits each train an infinite number
of times before it gets crushed. One could solve
the problem by summing an infinite geometric
series of distances.
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Geometric progression
A sequence is a geometric progression if the
ratio of any two successive terms is a fixed
number. a, a x, a x2, a x3,
We can easily find a formula for the sum of the
n1 first terms
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We can then try to add all infinite terms
?
n?8
If xgt1, then xn?8 and therefore the sum is
infinite. If xlt1, then xn?0 and therefore the
sum is equal to
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Koch snowflake
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Length of side l01
Number of sides a03
Number of triangles t01
Area of triangle A0 l02v3/4
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Length of side l11/3
Number of sides a134
Number of triangles t13
Area of triangle A1 l12v3/4
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Length of side l21/32
Number of sides a2a14
Number of triangles t2a1
Area of triangle A2 l22v3/4
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Length of side ln1/3n
Number of sides anan-14
Number of triangles tnan-1
Area of triangle An ln2v3/4
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The sequence of lengths ln is a geometric
progression.

• True
• False

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The sequence of numbers an is a geometric
progression.

• True
• False

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The sequence of numbers tn is a geometric
progression.

• True
• False

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The sequence of areas An is a geometric
progression.

• True
• False

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The total area enclosed by the snowflake is
__________. The total length of the curve that
makes the snowflake is ___________.

• Finite. Finite.
• Finite. Infinite.
• Infinite. Finite.
• Infinite. Infinite.

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length at the n-th step is (number of sides)
(length of side)
an ln
(3 4n ) (1/3n)
3 (4/3)n ?
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area at the n-th step is sum of areas for
various triangles
A0 t1 A1 tn-1 An-1
A0
S tk Ak The total area will be the limit of
the sum. However, this is a sum
of terms for a geometric
progression with
ratio x 4/9. (Explain this claim).
Therefore it
converges. In fact, show
that the total area is
( 2v3)/5
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Exponential Growth
A geometric progression increases very fast.
Lets try to get a feeling how fast the increase
is.
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Hundreds and hundreds of years ago there was a
King in India who loved to play games. But he had
gotten bored of the games that were present at
the time and wanted a new game that was much more
challenging. He commissioned a poor mathematician
who lived in his kingdom to come up with a new
game. After months of struggling with all kinds
of ideas the mathematician came up with the game
of Chaturanga. The game had two armies each lead
by a King who commanded the army to defeat the
other by capturing the enemy King. It was played
on a simple 88 square board.
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The King loved this game so much that he offered
to give the poor mathematician anything he wished
for. "I would like one grain of rice for the
first square of the board, two grains for the
second, four grains for the third and so on
doubled for each of the 64 squares of the game
board" said the mathematician. "Is that all?"
asked the King, "Why don't you ask for gold or
silver coins instead of rice grains". "The rice
should be sufficient for me" replied the
mathematician. The King ordered his staff to lay
down the grains of rice and soon learned that all
the wealth in his kingdom would not be enough to
buy the amount of rice needed
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22
23
1
2
27
25
26
24
214
212
210
28
211
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215
213
218
217
216
222
220
223
221
219
231
225
227
229
230
228
226
224
263
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How many grains of rice needed?
Assuming that a grain weighs about 1/10 gram,
this is more than 1,844 billion tons of
rice.
The world production of rice is about 560 million
tons per year. This means that, with todays
production, the King would need 3,293 years
to pay the debt.
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REVIEW QUESTION The numbers 0.3, -0.33,
0.333, -0.3333, 0.33333, -0.333333, are the
partial sums of a series that __________.

• True converges
• False does not converge
• True does not converge
• False converges

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We recognize the limit of the odd sums as the
number 0.33333333333333333333333333333333333333..
which is another way to express the
fraction 1/3 However the even sums converge
to -1/3 Since the two subsequences converge to
different limits, the original sequence does not
converge.
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REVIEW QUESTION If ,
are two geometric series, then is also a
geometric series.

• Always true.
• Never true.
• Sometimes true.
• There is not enough information to decide.

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REVIEW QUESTION Let ,
be two convergent series.
Which of the following statements is NOT true?
A. B. C. D.
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However
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John von Neumann
master of calculations
John von Neuman (1903-1957)
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THE PROBLEM
Two trains 200 miles apart are moving toward each
other each one is going at a speed of 50 miles
per hour. A fly starting on the front of one of
them flies back and forth between them at a rate
of 75 miles per hour. It does this until the
trains collide and crush the fly to death. What
is the total distance the fly has flown?
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How many grains of rice needed?
Assuming that a grain weighs about 1/10 gram,
this is more than 1,844 billion tons of
rice.
The world production of rice is about 560 million
tons per year. This means that, with todays
production, the King would need 3,293 years
to pay the debt.
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The previous story was used to teach us

• nothing. It was for our amusement.
• the definition of a geometric progression.
• the definition of a geometric series.
• how fast a geometric series increases.
• to do calculations with grains of rice.

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geometric/exponential increase
power increase
linear increase
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In the language of computers
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an application on ..
Vampires
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From Chapter 12 play clip
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The idea of whoever is bitten by vampire
becomes a vampire suffers from a fundamental
flaw Lets discuss this flaw more closely.
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(No Transcript)
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An estimation of the world population for any
date may be found at www.census.gov/ipc/www/world
his.html January 1, 1600 it is estimated that
the world population was (approximately)
536,870,000 humans
And, of course, at the same time we had
1 vampire.
What happened next?
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In February of 1600, we had 2 vampires
and (536,870,000 1) humans
In March of 1600, we had 4 vampires and
(536,870,000 3) humans
In April of 1600, we had 8 vampires and
(536,870,000 7) humans
In May of 1600, we had 16 vampires and
(536,870,000 15) humans
and so on
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22
23
1
2
27
25
26
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214
212
210
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211
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215
213
218
217
216
222
220
223
221
219
231
225
227
229
230
228
226
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and 268,435,001 humans
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If the first vampire had appeared in January 1,
1600 humans should have been wiped out by June 1,
1602. If we take into account newborn babies, we
could have survived an additional month
Apparently the result of this calculation
contradicts reality our existence. Therefore,
the assumption that the first vampire existed is
wrong vampires could not have existed
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• Proof by contradiction
• Arithmetic progression
• Geometric Progression
• Sum of an infinite series

Which of the ideas in the right box we used to
discredit the idea of vampires?

• Ideas 1 and 2
• Ideas 1 and 3
• Idea 1 only
• Idea 3 only
• Ideas 1, 3, and 4.

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the doubling of number we used above is a
geometric progression with ratio 2 and the proof
we presented is usually called reductio ad
absurdum (that is reduction to the absurd or
proof by contradiction). Besides mathematics, it
is been used very often in philosophy too (e.g.
Aristotle).
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In other words all this discussion reveals that
this idea of vampires could have originated only
from a person who flunked college algebra and
philosophy
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More serious applications.
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PROBLEM

• If a ball that is dropped from an initial height
  h bounces to ¼ of its previous height,
• how long does it take to come to rest?
• how far has the ball traveled if all heights are
  placed on a straight line?

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Before we answer it numerically, lets answer it
intuitively. T time to stop L distance
traveled

• T finite L finite
• T finite L infinite
• T infinite L finite
• T infinite L infinite

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Total distance h 2 h1 2 h2
2 (h/4) 2 (h/42)
2h (1/4) (1
1/4 ) 2h (1/4)
(4/3) 2h/3
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Total time t0 2 t1 2 t2
Times are found using the equation ½ g t2n
hn ? tn v(2hn/g)
2v(2/g) v h1 v h2
2v(2/g) v(h/4) v(h/42)
2v(2h/g) (1/2)
(1/22) 2v(2h/g)
(1/2) 1 (1/2)
2v(2h/g)
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Back to the trains and the fly. The solution
through series
vF
d1
v
v
D
The two trains are D miles apart at the time
the fly leaves the left train . The right train
and the fly will meet at point at distance d1
such that d1 vF t , D-d1 v t
Therefore D d1 d1 v/vF ? d1 D /
(1 v/vF)
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Notice that x 1/(1 v/vF) 1/(1
50/75) 3/5 We will also need y 2x -1
1/5 So d1 D x
While the fly was flying, the train from left was
moving. It traveled exactly D-d1 (i.e. the same
distance covered by the other train). Therefore,
when the fly reaches the other train the distance
between the trains is D1 D -2 (D-d1)
2d1-D (2x-1) D or D1 y D
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vF
d2
v
D1
v
Once the fly reaches the right train, the process
repeats itself with D1 as initial distance now
d2 D1 x D2 D1 y
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vF
d3
v
v
D2
Then the process repeats itself again from the
left d3 D2 x D3 D2
y then from the right, etc.
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Then we find dtotal d1 d2 d3
D x D xy D xy2
D x ( 1 y y2 ) D x /
(1-y) 200 miles (3/5) ( 5/4)
150 miles
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The Big Idea of Calculus
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What is the core idea of calculus in plain terms?

• Everything reduces to calculus.
• If you know calculus you can do science.
• When one wishes to compute a quantity of an
  object, he should split the object in small
  pieces and calculate the small quantities
  corresponding to the small pieces. Then, the
  total quantity is the sum of all small
  quantities.
• Infinite sums can be finite or infinite.
• Derivatives and integrals are the most
  fundamental objects of mathematics.

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The region under the curve is approximated with
rectangles.
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Finer partitions create more rectangles with
shorter bases.
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The solid sphere approximated with cross-section
based cylinders.
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Any object can be represented as an infinite sum
of infinitesimal pieces.
The infinitesimal pieces are really so small that
are effectively zero! Therefore, any finite
object is an infinite sum of zeros! This also
implies that an infinite sum is not necessarily
infinite. In many cases it gives a finite number.
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Zeno's Paradoxes
Zeno of Elea (c. 490 BC - 430 BC)
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Zeno of Elea was a philosopher. Unfortunately, we
know little of his life, and none of his
publications have survived.  We know of his work
only from other references.  Zeno did produce a
book that contained 40 paradoxes trying to argue
about the impossibility of motion. Motion is an
illusion and in reality it cannot even start.
Four of the paradoxes had significant effect on
mathematics. Aristotle's work refers to Zeno's
four paradoxes of "Dichotomy," "Achilles,"
"Arrow," and "Stadium."  Aristotle was not taken
in by Zeno's paradoxes and called them
fallacies.  However, it wasn't until more modern
times and through the development of the calculus
that mathematics developed the notation and
results to adequately handle Zeno's challenging
contradictions.
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• Zeno's paradoxes were a major problem for ancient
  and medieval philosophers.
• In modern times, calculus provides at least a
  practical solution (as we will see).
• Other proposed solutions to Zeno's paradoxes have
  included
• the denial that space and time are themselves
  infinitely divisible,
• the denial that the terms space and time refer
  to any entity with any innate properties at all.

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The Arrow and The Target
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TARGET
BEGIN
M1
M2
M3
And so onad infinitum.
Therefore, if the space is infinitely divisible,
the arrow can never reach the target. Even
worse, the motion cannot even start since one can
apply the same reasoning in the very first
interval.
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The answer of calculus is Zenos paradox assumes
that an infinite sum will always be infinite.
However, in many cases, an infinity of quantities
may be added to get a finite quantity.
Total distance L1 L2 L3
(L/2) (L/22) (L/23)
(L/2) (1 1/2
1/22 ) (L/2) 2
L
and in fact this will happen in a finite amount
of time. (Prove it.)
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Achilleas and Turtle
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The End