The power series of a function around a point \( t_0 \) is $$ g(t) = \sum_{n=0}^{\infty} \frac{(t-t_0)^n g^{(n)}(t_0)}{n!} .$$ Note that the \( g^{(n)}(t_0) \) denotes the derivative evaluated at \( t_0 \), where \( g^{(0)}(t_0) = g(t_0) \).
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Edit: Sorry, to be specific this is the Taylor power series.