1 Taylor series

Taylor series are power series generated from a function f

1.1 constructing a series

For x = a then a_n = \frac{f^{(n)}(a)}{n!}

Then if f has a series representation it must be:

f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots

1.2 Maclaurin series

If the series are centered around a = 0 we call the generated power series Maclaurin series.

1.3 order of n

Tells us about the ‘deepness’. How many derivatives we are taking (instead of abstract \infty). n is then called the order of the Taylor series.

P_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n

1.4 Taylor theorem

Let R_n be the remainder (error)

f(x) = P_n(x) + R_n(x)

Let I be an interval, x, a \in I there is a number c between x and a then R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1}

If \lim_{n \to \infty} R_n = 0 for all x \in I the Taylor series generated by f at x = a converges to f on I.