# 10.2 | Taylor Series

If $$f$$ has a power series representation (expansion) at $$c$$, that is, if
$$f(x) = \sum_{n = 0}^{\infty} a_n (x-c)^n \; \; |x-c| < R$$ then its coefficients are given by the formula $$a_n = \frac{f^{(n)}(c)}{n!}$$
The Taylor series of $$f(x)$$ centered at $$c$$ is given by
$$\sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-c)^n$$
That is,

$$f(x) = f(c) + f'(c)(x-c) + \frac{f^{\prime \prime} (c)}{2!} (x-c)^2 + \frac{f^{\prime \prime \prime} (c)}{3!} (x-c)^3 + \cdots$$
\begin{aligned} f(x) & = f(c) + f'(c)(x-c) + \frac{f^{\prime \prime} (c)}{2!} (x-c)^2 \\ & \; \; + \frac{f^{\prime \prime \prime} (c)}{3!} (x-c)^3 + \cdots \end{aligned}
\begin{aligned} f(x) & = f(c) + f'(c)(x-c) \\ & \; \; + \frac{f^{\prime \prime} (c)}{2!} (x-c)^2 \\ & \; \; + \frac{f^{\prime \prime \prime} (c)}{3!} (x-c)^3 + \cdots \end{aligned}

If $$c = 0$$, then this series is called a Maclaurin series.

The $$n$$-th degree Taylor polynomial of $$f(x)$$ centered at $$c$$ is given by
$$T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(c)}{k!}(x-c)^k$$

That is,

$$T_n(x) = f(c) + f'(c)(x-c) + \frac{f^{\prime \prime} (c)}{2!} (x-c)^2 + \cdots + \frac{f ^{(n)} (c)}{n!} (x-c)^n$$
\begin{aligned} T_n(x) & = f(c) + f'(c)(x-c) \\ & \; \;+ \frac{f^{\prime \prime} (c)}{2!} (x-c)^2 + \cdots \\ & \; \;+ \frac{f ^{(n)} (c)}{n!} (x-c)^n \end{aligned}
\begin{aligned} T_n(x) & = f(c) + f'(c)(x-c) \\ & \; \;+ \frac{f^{\prime \prime} (c)}{2!} (x-c)^2 + \cdots \\ & \; \;+ \frac{f ^{(n)} (c)}{n!} (x-c)^n \end{aligned}

The remainder is given by $$R_n(x) = f(x) – T_n(x)$$
If $$c = 0$$, then this series is called a Maclaurin polynomial and we use the notation $$M_n(x)$$ instead of $$T_n(x)$$.