MATH 136 -- AP Calculus
November 24, 2003
Polynomials approximating
From the geometric series sum formula we know
that + ... 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]); |
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]); |
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]); |
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.