Keywords: Taylor’s Theorem, Absolute Convergence
This is the Chapter10 ReadingNotes from book Thomas Calculus 14th.
Taylor and Maclaurin Series（泰勒和麦克劳林级数）
Series Representations(级数表示)
Taylor and Maclaurin Series
DEFINITIONS
Let $ƒ$ be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by $ƒ$ at $x = a$ is
$$
\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k = f(a) + f’(a)(x-a) + \frac{f’’(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \cdots
$$
The Maclaurin series of $ƒ$ is the Taylor series generated by $ƒ$ at $x = 0$, or
$$
\sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}(x-0)^k = f(0) + f’(0)(x-0) + \frac{f’’(0)}{2!}(x-0)^2 + \cdots + \frac{f^{(n)}(0)}{n!}(x-0)^n + \cdots
$$
Taylor Polynomials
DEFINITION
Let $ƒ$ be a function with derivatives of order $k$ for $k = 1, 2, . . . , N$ in some interval containing $a$ as an interior point. Then for any integer $n$ from $0$ through $N$, the Taylor polynomial of order $n$ generated by $ƒ$ at $x = a$ is the polynomial
$$
P_n(x) = f(a) + f’(a)(x-a) + \frac{f’’(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(k)}(a)}{k!}(x-a)^k + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n
$$
Just as the linearization of $ƒ$ at $x = a$ provides the best linear approximation of $ƒ$ in the neighborhood of $a$, the higher-order Taylor polynomials provide the “best” polynomial approximations of their respective degrees.
Convergence of Taylor Series
THEOREM 23—Taylor’s Theorem
If $ƒ$ and its first $n$ derivatives $ƒ’, ƒ’’, \cdots, ƒ^{(n)}$ are continuous on the closed interval between $a$ and $b$, and $ƒ^{(n)}$ is differentiable on the open interval between $a$ and $b$, then there exists a number $c$ between $a$ and $b$ such that
$$
f(b) = f(a) + f’(a)(b-a) + \frac{f’’(a)}{2!}(b-a)^2 + \cdots + \frac{f^{n}(a)}{n!}(b-a)^n + \frac{f^{n+1}(c)}{(n+1)!}(b-a)^{n+1}
$$
Taylor’s Theorem is a generalization of the Mean Value Theorem.
Taylor’s Formula
If $ƒ$ has derivatives of all orders in an open interval $I$ containing $a$, then for each positive integer $n$ and for each $x$ in $I$,
$$
f(x) = f(a) + f’(a)(x-a) + \frac{f’’(a)}{2!}(x-a)^2 + \cdots + \frac{f^n(a)}{n!}(x-a)^n + R_n(x)
\tag{1}
$$
where
$$
R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}
\tag{2}
$$
for some $c$ between $a$ and $x$
Equation (1) is called Taylor’s formula. The function $R_n(x)$ is called the remainder of order $n$ or the error term for the approximation of $ƒ$ by $P_n(x)$ over $I$.
If $R_n(x) \rightarrow 0$ as $n \rightarrow \infty$ for all $x \in I$, we say that Taylor series generated by $f$ at $x = a$ converges to $f$ on $I$, and we write,
$$
f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k
$$
Estimating the Remainder
THEOREM 24—The Remainder Estimation Theorem
If there is a positive constant $M$ such that $|ƒ^{(n+1)}(t)| \leq M$ for all $t$ between $x$ and $a$, inclusive, then the remainder term $R_n(x)$ in Taylor’s Theorem satisfies the inequality
$$
|R_n(x)| \leq M\frac{|x-a|^{n+1}}{(n+1)!}
$$
If this inequality holds for every $n$ and the other conditions of Taylor’s Theorem are satisfied by $ƒ$, then the series converges to $ƒ(x)$.
