According to this post, approximation of $e^x$ by continued fractions is faster than Taylor series.
Generally speaking, are continued fractions good approximation method?
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Continued fractions yield the best approximations to numbers. Here is s posting with more details – Mittens Jan 01 '23 at 02:08
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If you can derive such an approximation, then yes, they will be very good. Continued fraction expansions can be viewed as a family of rational functions, so at the very least, they are a larger function space than polynomials (e.g. Taylor approximation), so at the outset, so their ability to approximate different kinds of functions will be better.
One huge advantage of rational functions is that they can uniformly approximate meromorphic functions, according to Runge's theorem. That is, you can use them to approximate functions that have singularities of a certain form.
Optimal rational function approximations, which is closely connected with continued fractions, are often called Pade approximants.
Christopher A. Wong
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