Usually, when trying to evaluate e^(-x), its Taylor series is used.
This is e^(-x) = Sum over n from 0 to infinity of (-x)^n/n! .
However, when x>2, many Taylor series terms are needed to approximate e^(-x) accurately.
Is there a more concise approximation to e^(-x) that is accurate for larger x values,
and only involves powers of x?
Thanks for your assistance in answering this question.
Check this out
https://www.researchgate.net/post/Are_there_any_approximation_methods_better_than_Taylor_approximation_for_exponential_functions_with_complex_number_in_the_exponent#:~:text=You%20are%20taking%20about%20complex,both%20negative%20and%20positive%20powers.
Dear Ron, since polynomials always diverge for $x\to\pm\infty$, any polynomial approximation of a bounded function can only have a limited interval of validity - the crucial information is which interval of the X-axis you focus on. However, polynomial approximations better than Taylor's may be obtained, for example, with interpolating polynomials constructed on Chebyshev nodes.
If you can move to rational approximations, a better alternative would be given by Padé approximants, which may be used on unbounded intervals:
https://en.wikipedia.org/wiki/Pad%C3%A9_approximant
There are countless possible approximations. The accuracy of these depends on their formulation and also the interval (a≤x≤b) over which they are used. The bigger question is what do you want to do and why do you need an approximation for exp(-x)? Without knowing this we can't provide a helpful answer.
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