Kalkulus II (09 - 10)

Start with a square one unit by one unit:1This is an example of an infinite series.1This series converges (approaches a limiting value.)Many series do not converge:1/9/20114© 2010 Universitas Negeri Jakarta | www.unj.ac.id |

If Sn has a limit as , then the series converges, otherwise it diverges.In an infinite series:a1, a2,… are terms of the series.an is the nth term.Partial sums:nth partial sum1/9/20115© 2010 Universitas Negeri Jakarta | www.unj.ac.id |

is the interval of convergence.Geometric Series:In a geometric series, each term is found by multiplying the preceding term by the same number, r.1/9/20116© 2010 Universitas Negeri Jakarta | www.unj.ac.id |

A power series is in this form:orThe coefficientsc0, c1, c2… are constants.The center “a” is also a constant.(The first series would be centered at the origin if you graphed it. The second series would be shifted left or right. “a” is the new center.)1/9/201110© 2010 Universitas Negeri Jakarta | www.unj.ac.id |

Example 3:multiply both sides by x.To find a series forOnce we have a series that we know, we can find a new series by doing the same thing to the left and right hand sides of the equation.This is a geometric series where r=-x.1/9/201111© 2010 Universitas Negeri Jakarta | www.unj.ac.id |

The previous examples of infinite series approximated simple functions such as or .This series would allow us to calculate a transcendental function to as much accuracy as we like using only pencil and paper!1/9/201115© 2010 Universitas Negeri Jakarta | www.unj.ac.id |

Maclaurin Series:(generated by f at )Taylor Series:(generated by f at )If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series:1/9/201122© 2010 Universitas Negeri Jakarta | www.unj.ac.id |

Maclaurin Series:(generated by f at )MaclaurinSeriesThere are some Maclaurin series that occur often enough that they should be memorized. They are on your formula sheet, but today we are going to look at where they come from.1/9/2011© 2010 Universitas Negeri Jakarta | www.unj.ac.id |27

We wouldn’t expect to use the previous two series to evaluate the functions, since we can evaluate the functions directly.They do help to explain where the formula for the sum of an infinite geometric comes from.We will find other uses for these series, as well.A more impressive use of Taylor series is to evaluate transcendental functions.1/9/201132© 2010 Universitas Negeri Jakarta | www.unj.ac.id |

and substitute for , we get:If we start with this function:This is a geometric series with a = 1 and r = -x2.If we integrate both sides:This looks the same as the series for sin (x), but without the factorials.1/9/201135© 2010 Universitas Negeri Jakarta | www.unj.ac.id |

Kalkulus II (09 - 10)

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Kalkulus II (09 - 10)