• Taylor Polynomial
Given a smooth function f (x). The Taylor Series expansion for f is
∑∞ k=0
f(k)(c)
k! (x− c)k
The Taylor Polynomial of degree n, denoted by Pn(x), expanded about x = c is given by
Pn(x) =
∑n k=0
f(k)(c)
k! (x− c)k
• Taylor’s Theorem Suppose that f has derivatives of all orders in the in- terval (c− r, c + r), for some r > 0 and that lim
n→∞Rn(x) = 0, for all x in (c− r, c + r). Then, the Taylor series for f expanded about x = c converges to f (x), that is,
f (x) =
∑∞ k=0
f(k)(c)
k! (x− c)k, for all x in (c− r, c + r).
