# Calc09_3 - 9.3 Taylors Theorem Error Analysis for Series...

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9.3 Taylor’s Theorem: Error Analysis for Series Tacoma Narrows Bridge: November 7, 1940 Greg Kelly, Hanford High School, Richland, Washington

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Taylor series are used to estimate the value of functions (at least theoretically - now days we can usually use the calculator or computer to calculate directly.) An estimate is only useful if we have an idea of how accurate the estimate is. When we use part of a Taylor series to estimate the value of a function, the end of the series that we do not use is called the remainder. If we know the size of the remainder, then we know how close our estimate is.
For a geometric series, this is easy: ex. 2: Use to approximate over . 2 4 6 1 x x x + + + 2 1 1 x - ( 29 1,1 - Since the truncated part of the series is: , 8 10 12 x x x + + + ⋅⋅⋅ the truncation error is , which is . 8 10 12 x x x + + + ⋅⋅⋅ 8 2 1 x x - When you “truncate” a number, you drop off the end. Of course this is also trivial, because we have a formula that allows us to calculate the sum of a geometric series directly.

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Taylor’s Theorem with Remainder If f has derivatives of all orders in an open interval I containing a , then for each positive integer n and for each
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Calc09_3 - 9.3 Taylors Theorem Error Analysis for Series...

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