Infinity and Beyond!

A prelude to Infinite Sequences and Series (Ch 12)

Infinity and Fractals…

• Fractals are self-similar objects whose overall geometric form and structure repeat at various scales they provide us with a “glimpse” into the wonderful way in which nature and mathematics meet.

• Fractals often arise when investigating numerical solutions of differential (and other equations).

• Fractals provide a visual representation of many of the key ideas of infinite sequences and series.

Paradoxes of Infinity

• Zeno– Motion is impossible– Achilles and the

tortoise– Math prof version

The Koch Snowflake and Infinite Sequences…

What is a Koch Snowflake?• How “long” is a section of

the Koch Snowflake between x = 0 and x = 1?

• Anything else odd about this?– What “dimension” is it?– Can you differentiate it?

What is the area of a Koch Snowflake?• Start with this…

3

4

2 23 1 3 1

34 3 4 3

43 112 ( )

4 32 3

3 5 7

3 1 4 4 4(1 )

4 3 3 3 3

Rules of the Game…

• Section 12.1 – defines sequence and basic terminology

• Section 12.2 – extends definitions to infinite series

• Use many of the ideas that you developed about limits in Math 200 and 205

• Important Theorems:– The Squeeze Theorem– L’Hopital’s Rules Examples: pg 747-48: 5, 12, 33

Convergence

• True or Falsea series for which

must converge.

lim 0nna

Examples: 756-57: 2, 21,44