Some Important Power Series (4/3/06)

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Some Important Power Series (4/3/06)

• We shall see in upcoming sections that certain
  power series, called Taylor Series, can be used
  to express exactly non-algebraic functions like
  ex, ln(x), sin(x), etc., or to approximate them
  as closely as we like.
• This is how your hand calculator computes these
  functions it uses the first so-many terms of
  certain infinite series.

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A simple (but nice) example

• Consider the series1 x x2 / 2! x3 / 3!
  xn / n!
• What is its interval of convergence?
• Lets plot the first few terms of this function
  and see what it looks like.
• Take the derivative term by term of this
  function. What do you get? Sound familiar?

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Another (nice) example

• Take every other term of the previous series
  (only odd n) and alternate the signs of the
  terms x x3 / 3! x5 / 5! (-1)n1x2n-1/
  (2n-1)!
• Plot some terms of this. What does it look like?
• Again, take the derivative. Want to guess what
  that is?

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And yet another

• Here is one which is centered at x 1(x-1)
  (x-1)2 / 2 (x-1)3 / 3 - (x-1)4 / 4
• Whats the interval of convergence here?
• Again, lets plot some terms Whats this one
  look like?
• Again, take the derivative. Note that the answer
  is geometric!! Whats its sum? Wow!

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Comment and Assignment

• We shall see later this week how to construct
  these very useful power series.
• Until thenFor Wednesday We shall first meet in
  the classroom, then move to the lab for Lab
  6.Also, hand-in assignment 3 is due at 5 pm
  (so that you can ask last-minute questions in
  class).