Taylor’s Formula Taylor’s formula describes how to get power series representations of functions. The function ex doesn’t look like a polynomial; we have to ﬁgure out what the values of ai have to be in order to describe ex as a series. Taylor’s formula says that given any function f for which the nth derivative (n) f (x) exists for x near 0, f (x) = ∞ � f (n) (0) n x . n! n=0 We’ll learn how to use it soon. Why should this work? Suppose that: f (x) = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · Then: f � (x) = a1 + 2a2 x + 3a3 x2 + 4a4 x3 + · · · �� 2 f (x) = 2a2 + 3 · 2a3 x + 4 · 3a4 x + · · · and and f (3) (x) = 3 · 2a3 + 4 · 3 · 2a4 x + · · · Evaluating each of these at 0 we see that: f (0) = a0 , f � (0) = a1 , f �� (0) = 2a2 f (3) (0) and in general: and f (3) (0) = 3 · 2a3 . Solving for a3 we get a3 = 3·2·1 an = f (n) (0) , n! where: n! = n · (n − 1) · (n − 2) · · · 1. We deﬁne 0! = 1 because that makes our formulas work nicely.

Taylor’s Formula

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