A Geometric Series

 In calculus, you will learn that certain functions can be approximated by polynomial functions. We will explore one such function now. 

(a) Using a graphing utility, create a table of values with 

\( Y _ { 1 } = f ( x ) = \frac { 1 } { 1 - x } \) and \( Y _ { 2 } = g _ { 2 } ( x ) = 1 + x + x ^ { 2 } + x ^ { 3 } \) 

for \( - 1 < x < 1 \) with \( \Delta \) Tbl \( = 0.1 \) . 

(b) Using a graphing utility, create a table of values with 

\( Y _ { 1 } = f ( x ) = \frac { 1 } { 1 - x } \) and 

\( Y _ { 2 } = g _ { 3 } ( x ) = 1 + x + x ^ { 2 } + x ^ { 3 } + x ^ { 4 } \) 

for \( - 1 < x < 1 \) with \( \Delta \) Tbl \( = 0.1 \) . 

(c) Using a graphing utility, create a table of values with 

\( Y _ { 1 } = f ( x ) = \frac { 1 } { 1 - x } \) and 

\( Y _ { 2 } = g _ { 4 } ( x ) = 1 + x + x ^ { 2 } + x ^ { 3 } + x ^ { 4 } + x ^ { 5 } \) 

for \( - 1 < x < 1 \) with \( \Delta \) Tbl \( = 0.1 \) . 

(d) What do you notice about the values of the function as more terms are added to the polynomial? Are there some values of \( x \) for which the approximations are better?