Intro to Sequences and Series. They are enjoying the camp fire and the turtle starts to tell a...

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..........32

1,

16

1,

8

1,

4

1,

2

1

kka

a

a

a

a

2

1

16

18

14

12

1

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 2

1

2nd 4

1

8

1

16

1

General term

.........32

1

16

1

8

1

4

1

2

1

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 2

1

kk

Would this ever end? Namely does this sum has a finite value?

2

1

4

1

8

1

16

1

32

1

64

1

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 2

1

2

1)(

dx

xf

x

x

What do you know about the integral ?

Is it convergent or divergent?

So, it is convergent, namely

0 2

1dx

x

2ln

1

2ln

1

2)2(ln

1lim

2)2(ln

1lim

2

1lim

2

10

00

tt

t

xt

t

xtxdxdx

2ln

1

2

1

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 .

x2

1

2ln

1

2ln

1........

16

1

8

1

4

1

2

1

2ln

11

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