In general, any function whose derivatives of all orders are
finite can be written as a power series
|
f(x) = f(0) + x f¢(0) + |
x2
2!
|
f¢¢(0) + ... + |
xn
n!
|
f(n)(0)+... |
|
This expansion is an infinite series (not a polynomial).
Truncating this series at a given point provides us with a
polynomial approximation to f(x).
The question of how big the errors are in this approximation is
a difficult one to answer, and more details will be discovered
at university in Numerical Analysis courses.