Taylor and Maclaurin Series

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The Formula for Taylor Series

A function is called analytic if it can be expressed as an infinite power series around some point $a$ . Taylor's theorem gives us a formula for the coefficients of the power series expansion of an analytic function:

Taylor's Theorem

If

$\displaystyle f(x) = \sum_{n=0}^\infty c_n (x-a)^n$ , then

$\displaystyle c_n = \frac{f^{(n)}(a)}{n!}$ .

This formula works both ways: if we know the $n$ -th derivative, we can figure out $c_n$ ; if we know $c_n$ , we can figure out the $n$ -th derivative.