Intro to Sequences and Series

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!!!

Zzzzzzz!

The duckling is to tired to listen to the whole story and falls asleep!!!

Shoot!Zeno’s

paradox

1 km

1/2

1/4

1/8

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

.........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?

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…

• 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

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

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

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

Calculus isawesome!

I am happy!

THE END