intro to sequences and series
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Intro to Sequences and Series. One day they decide to go camping in FarmVille !!!. They are enjoying the camp fire and the turtle starts to tell a story. He says:. “This is a real story about my great great great great great great ….. grandfather…….This is called Zeno’s paradox. - PowerPoint PPT PresentationTRANSCRIPT
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Intro to Sequences and Series
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They are enjoying the campfire and the turtle startsto tell a story. He says:
“This is a real story about my great great great great great great …..grandfather…….This is called Zeno’s paradox.……..”
One day they decide to go camping in FarmVille!!!
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Zzzzzzz!
The duckling is to tired to listen to the whole story and falls asleep!!!
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Shoot!Zeno’s
paradox
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1 km
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1/2
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1/4
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1/8
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This is called a sequence. Informally a sequence is an infinite list..........321,
161,
81,
41,
21
kka
a
a
a
a
21
161
814121
4
3
2
1
What is a sequence of real numbers?More formally…
Input OutputA sequence of realnumbers is a functionin which the inputs are positive integers and the 3rd outputs are real numbers. 4th
1st 21
2nd 41
81
161
General term
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.........321
161
81
41
21
I have to walk all these pieces, but…….
To save some time how can I write this sum?
This is called an infinite series.
1 21
kk
Would this ever end? Namely does this sum has a finite value?
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21
41
81
161
321
641
1 1 1 1 11…………….
Geometrically…
+ + + …………….+
To find the total distance that the duckling needs to walk, we add up all the areas…
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• What are these rectangles trying to do? Riemann approximation
• For which integrand? For which integral?
• Is this approximation an over or underestimate? Underestimate
0 21
21)(
dx
xf
x
x
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What do you know about the integral ?
Is it convergent or divergent?
So, it is convergent, namely
0 21 dxx
2ln1
2ln1
2)2(ln1lim
2)2(ln1lim
21lim
21
000
tt
txt
t
xtx dxdx
2ln1
21
0
dxx
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Conclusion: Since the sum of the areas of the rectangles are smaller than the area A below the graph of , these areas add up to a finite number that is less than .
x21
2ln1
2ln1........
161
81
41
21
2ln11
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The concepts that the duckling has learned:
• Sequences
A general sequence can be written more compactly as
•Infinite series
•How they can be connected to integrals, convergence, divergence ideas…
•Don’t mess with infinity!!!
,....,,, 4321 aaaa
k=1 or simply .k ka a
1kka
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Calculus isawesome!
I am happy!
THE END