MATH 136 -- AP Calculus

November 24, 2003

Polynomials approximating   f(x) = 1/(1-x)

From the geometric series sum formula we know

that 1/(1-x) = 1+x+x^2+x^3  + ...  for  -1 < x  < 1

Here is some additional graphical evidence:

>    plot([1/(1-x), 1, 1+x, 1+x+x^2, 1+x+x^2+x^3,1+x+x^2+x^3+x^4],x=-1..1,y=0..5,color=[red,black,blue,gray,cyan,green]);

[Maple Plot]

Note that as we increase the degree of the polynomial, the approximation gets better,

and also stays better on a larger subinterval of (-1,1).  For instance, the second degree

polyomial:

>    plot([1/(1-x), 1+x+x^2],x=-1..1,y=0..5,color=[red,gray]);

[Maple Plot]

is only really close between about x = -0.3  and x = +0.3.    On the other hand, the

4th degree polynomial is

>    plot([1/(1-x), 1+x+x^2+x^3+x^4],x=-1..1,y=0..5,color=[red,green]);

[Maple Plot]

is close enough to the graph that the difference is not visible at this graphics

resolution level from about   x = -0.5  to  x = +0.5.