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…

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 False a series for which must converge. Examples: 756-57: 2, 21,44