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Accuracy of series approximations

 
 
Levi
 
Reply Sun 1 Apr, 2007 10:06 am
How would I estimate a range of x values for which a series approximation is accurate to within a certain error?

For example, in the (I guess Maclaurin) series expansion for sin(x), if I take the expansion out to n=1,
sin(x) = x - ((x^3)/6)
(I don't know how to make it a squiggly equals sign)
and I want to the error to be less than .01, how do I find the range of x values for which that's true?
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stuh505
 
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Reply Sun 1 Apr, 2007 02:17 pm
simply plug into the definition of error and find the range where it is less than 0.01

relative error = |x - ((x^3)/6) - sin(x)| / sin(x)
absolute error = |x - ((x^3)/6) - sin(x)|

in the case of relative error that comes out to be |x| < 1.0077925
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raprap
 
  1  
Reply Sun 1 Apr, 2007 06:06 pm
An easy way is to test the nth expansion factor. On convergent series like Maclaurin series, if the nth factor is within error limits, the series expansion is acceptable.

Generally, though, I go one more term to salve my monkey counting gene.

Rap
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