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I am looking for a series expansion which will converge faster than the Taylor series. I mean $$ f(x)=\sum_{n=0}^{N}\frac{f^{(n)}(0)}{n!}x^n $$

For some function you may need large $N$ to get a convergent series. Is there a faster convergent series?

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    $\begingroup$ depends on the function. $\endgroup$ – Will Jagy Apr 20 '16 at 1:50
  • $\begingroup$ For some functions, its Taylor series won't converge to the function at more than one point $\endgroup$ – Rick Sanchez Apr 20 '16 at 1:52
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    $\begingroup$ the series $f,0,0,0\dots$ converges really fast. $\endgroup$ – Jorge Fernández Hidalgo Apr 20 '16 at 1:54
  • $\begingroup$ @CarryonSmiling haha $\endgroup$ – MathMajor Apr 20 '16 at 1:59
  • $\begingroup$ some times one can re-cast the Taylor to a sum of Chebyshev polynomials: en.wikipedia.org/wiki/Chebyshev_polynomials... $\endgroup$ – Chip Apr 20 '16 at 2:11

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