# Is there a faster convergence series than the Taylor series?

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?

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