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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.
……..”
The duckling is to tired to listen to the whole story and falls asleep!!!
Zzzzzzz!
Shoot!
Zeno’s
paradox
1 km
1/2
1/4
1/8
1 1 1 1 1
, , , , .........
2 4 8 16 32
1
2
1
a2 
4
1
a3 
8
1
a4 
16
a1 

General term
ak 
1
2k

This is called a sequence.
Informally a sequence is an infinite list.
What is a sequence of real numbers?
More formally…
Input
Output
1
A sequence of real
1st
2
numbers is a function
nd
1
2
in which the inputs are
4
1
rd
positive integers and the
3
8
outputs are real numbers.
1
4th
16


I have to walk all these pieces, but…….
1 1 1 1 1
     .........
2 4 8 16 32
This is called an infinite series.
To save some time how can I write this sum?

1

k
k 1 2
Would this ever end? Namely does this sum has a finite value?
Geometrically…
1
2
1
4
1
1
8
1
16
1
1
1
32
1
…………….
1
64
1
1
To find the total distance that the duckling needs to walk, we add up all the areas…
+
+
+
+
…………….
• What are these rectangles trying to do?
Riemann approximation
• For which integrand? For which integral?
f ( x) 


0
1
2x
1
dx
2x
• Is this approximation an over or underestimate?
Underestimate

What do you know about the integral
1
0 2 x dx ?
Is it convergent or divergent?

t
 1
1
1

dx

lim
dx

lim
0 2 x t  0 2 x
t   (ln 2)2 x

So, it is convergent, namely

1
1
0 2 x dx  ln 2

 1
1 
1




lim


 ln 2
0
t   (ln 2)2t
ln
2



t
Conclusion: Since the sum of the areas of the rectangles are smaller than the area A
below the graph of
than
1
ln 2
1
2x
, these areas add up to a finite number that is less
.
1 1 1 1
1
    ........ 
2 4 8 16
ln 2
1
1
ln 2
The concepts that the duckling has learned:
• Sequences
A general sequence a1 , a2 , a3 , a4 ,.... can be written more compactly as
ak k=1

or simply ak  .
•Infinite series

a
k 1
k
•How they can be connected to integrals, convergence, divergence ideas…
•Don’t mess with infinity!!!
THE END
Calculus is
awesome!
I am
happy!