Infinity and Beyond! A prelude to Infinite Sequences and Series (Ch 12)
Jan 06, 2016
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