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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?
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
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.
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