Jan 14, 2002
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- Intro to 3414
	- what a wonderful course this is! :)
	- applications in
		- Google
		- Microsoft Word (drawing)
		- Speedometers and Odometers
		- In short, they are everywhere!

- Basic approach
	- "how to do mathematics by computer"
	- why is this a big deal?
		- "computers are stupid"

- Classic example 
	- a = 4./3.;
	  b = a - 1.;
	  c = b + b + b;
	  print (c - 1);

- Review of Taylor's series
	- 1+2+..+k is unbounded
	- so is 1+1/2+1/3+..+1/k ! (even though each number
	  is smaller than the previous)

- How do you slightly modify the second to get a bounded series?
	- use alternating signs
		- 1-1/2+1/3-1/4... => ln 2
	- use factorials
		- 1/1!+1/2!+1/3!+... => e-1
	- use squared entries
		- 1+1/4+1/9+1/16+.. => ???

- First two are well known examples of Taylor's series expansions
	- first works when |x| < inf
	- second works in -1 < x <= 1

- Review Taylor's series expansion

- When does the series represent the function?
	- Taylor's theorem tells us this

- Examples
	- f(x) = 2x^2 + 3x + 1
	- f(x) = sin x

- In general, Taylor's series is good for approximating f() near
  the point of expansion (i.e., c)

- What happens if we move away from c?

- Alternating Series Theorem
	- Worked out example 8 from your textbook