Answer to Question #177374 in Calculus for Yusuf Idris

Question

Obtain a four-term Taylor polynomial approximation valid near x = 0 for each (1+ x)-⅓. Estimate the range of x over which three term polynomials will give two decimal accuracy.

Answer 1

Given, expression is-

$f(x)=(1+x)^{-\frac{1}{3}}$

On expanding the expression, we get the four term Taylor polyomial

$f(x)=1-\dfrac{1}{3}x+\dfrac{2}{9}x^2-\dfrac{14}{81}x^3$

Range i.e. Region of convergence of x is |x|<1