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Calc09_2day2 - 9.2 day 2 Maclaurin Series Liberty Bell...

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9.2 day 2 Maclaurin Series Liberty Bell, Philadelphia, PA Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003
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There 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. Maclaurin Series: (generated by f at ) 0 x = ( 29 ( 29 ( 29 ( 29 ( 29 2 3 0 0 0 0 2! 3! f f P x f f x x x ′′ ′′′ = + + + + ⋅⋅⋅
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( 29 ( 29 ( 29 ( 29 ( 29 2 3 0 0 0 0 2! 3! f f P x f f x x x ′′ ′′′ = + + + + ⋅⋅⋅ ( 29 1 1 1 1 x x - = - - ( 29 1 1 x - - ( 29 2 1 x - - ( 29 3 2 1 x - - ( 29 4 6 1 x - - ( 29 5 24 1 x - - ( 29 ( 29 n f x List the function and its derivatives.
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List the function and its derivatives. Evaluate column one for x = 0 . ( 29 ( 29 ( 29 ( 29 ( 29 2 3 0 0 0 0 2! 3! f f P x f f x x x ′′ ′′′ = + + + + ⋅⋅⋅ ( 29 1 1 1 1 x x - = - - ( 29 1 1 x - - ( 29 2 1 x - - ( 29 3 2 1 x - - ( 29 4 6 1 x - - ( 29 5 24 1 x - - 1 1 2 6 3! = 24 4! = ( 29 ( 29 0 n f ( 29 ( 29 n f x 2 3 4 2 3 1 1 2! 3! ! 4! 1 ! 1 4 x x x x x = + + + + +⋅⋅⋅ - 2 3 4 1 1 1 x x x x x = + + + + +⋅⋅⋅ - This is a geometric series with a = 1 and r = x .
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( 29 1 1 x - 1 1 x - x x + 2 x x - 2 x + 2 x 2 3 x x - 3 x 3 x + +⋅⋅⋅ We could generate this same series for with polynomial long division : 1 1 x -
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