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• Taylor Polynomial Given a smooth function f (x). The Taylor Series expansion for f is ∞∑ k

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• 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).

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