A Comparison of Some Taylor and Chebyshev Series

Abstract

A function is approximated in the interval $- 1 \leq x \leq 1$ by (i) a Taylor series in x; (ii) a Taylor series in $y = (x + \lambda )/(1 + \lambda x)$; (iii) a Chebyshev series in x; and (iv) a Chebyshev series in $z = (x + \mu )/(1 + \mu x)$. The convergence of all four series is discussed, and a method is given for finding the values of $\lambda$ and $\mu$ which optimize convergence. Methods are also given for transforming one of the above series into another, some of which provide effective methods for acceleration of convergence. The application of the theory to even and odd functions is also discussed.