Chapter 4.2, Problem 47AYU

Challenge Problem A Geometric Series In calculus you will learn that certain functions can be approximated by polynomial functions. We will explore one such function now.

Using graphing utility, create a table of values with

$Y_1=f\left(x\right)=\frac{1}{1-x}$ and $Y_2=g_2\left(x\right)=1+x+x^2+x^3$ for $-1<x<1$ and $\Delta \text{Tbl}\text{=}\text{0}\text{.1}$.

Using graphing utility, create a table of values with

$Y_1=f\left(x\right)=\frac{1}{1-x}$ and $Y_2=g_3\left(x\right)=1+x+x^2+x^3+x^4$ for $-1<x<1$ and $\Delta \text{Tbl}\text{=}\text{0}\text{.1}$.

Using graphing utility, create a table of values with

$Y_1=f\left(x\right)=\frac{1}{1-x}$ and $Y_2=g_4\left(x\right)=1+x+x^2+x^3+x^4+x^5$ for $-1<x<1$ and $\Delta \text{Tbl}\text{=}\text{0}\text{.1}$.

What do you notice about the values of the functions as more terms are added to the polynomial? Are there some values of $x$ for which the approximations are better?