7.6 Series

Approximating a function by polynomials is quite useful and leads naturally to the theory of infinite series. Suppose we are considering a function f( x) and would like to find a polynomial that closely approximates f( x) over some domain. Of course, if f itself is already a polynomial, then there is not much to do! So we should be thinking that f is not a polynomial. Perhaps f is a trigonometric function, or a logarithm or some other interesting function. Finding a polynomial that closely approximates f will be useful because polynomials are easy to compute and work with.

What does it mean for a polynomial P( x) to approximate f( x)? It is probably unrealistic to hope that P will be close to f for all values of x, so we mean that near some point the two functions are nearly equal. So, let's pick a point, say x = a, and try to approximate f near a. Since we want the functions to be nearly the same near a, it makes sense to demand that P and f are exactly the same at a, that is P( a) = f( a). This means that the two graphs will intersect at x = a. But we want the graphs to be nearly the same, so let's make...