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 \(T_n(x)\).

E 10.2 Exercises