SciTech-Mathmatics-Taylor Equation泰勒公式: 用幂级数(幂函数的和) 去无穷拟合 N阶可导函数(连续可导有N阶导数)

https://www.mathsisfun.com/algebra/taylor-series.html
https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/08%3A_Sequences_and_Series/8.08%3A_Taylor_Series

Derivative and Slope

Quick review: a $derivative$ gives us the $\text{slope of a function}$ at $anypoint$.
These derivative rules can help us:

• The derivative of $aconstant$ is 0
• The derivative of $ax$ is $a$ (example: the derivative of $2x$ is $2$)
• The derivative of $x^n$ is $n{x}^{n-1}$ (example: $\text{the derivative of }{x}^3\text{ is }3x^2$)
• We will use the little mark ${}^{\prime }$ to denote "$\text{derivative of}$" (example: $f^{\prime }(x)$ denote the $\text{derivative of }f(x)$).

Taylor Series:

• $\text{First define }{f}^{(0)}(x)=f(x)\text{ and }0!=1$ :

• Formula:
  $\begin{array}{r}\\ f(x)& =& \sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n\\ & =& f(x_0)+\sum _{n=1}^{\mathrm{\infty }}\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n\end{array}$

• Examples:
  $\begin{array}{r}\\ e^x& =& \sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}\\ & =& 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\cdots +\frac{x^n}{n!}\\ \cos (x)& =& \sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^{(n)}}{(2n)!}{x}^{(2n)}\\ & =& 1+\frac{(-1)x^2}{2!}+\frac{(+1)x^4}{4!}+\cdots +\frac{(-1{)}^{(n)}}{(2n)!}{x}^{(2n)}\\ \sin (x)& =& \sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1)!}{x}^{(2n+1)}\\ & =& x+\frac{(-1)x^3}{3!}+\frac{(+1)x^5}{5!}+\cdots +\frac{(-1{)}^n}{(2n+1)!}{x}^{(2n+1)}\end{array}$
  Let $x=i\cdot y$ and $i^2=-1$:
  $\begin{array}{r}\\ e^x=e^{i\cdot y}& =& \sum _{n=0}^{\mathrm{\infty }}\frac{(i\cdot y{)}^n}{n!}\\ & =& 1+i\cdot y+\frac{-1\cdot y^2}{2!}+i\cdot \frac{-1\cdot y^3}{3!}+\frac{+1\cdot y^4}{4!}+i\cdot \frac{+1\cdot y^5}{5!}+\cdots +i^n\cdot \frac{(y^n}{n!}\\ & =& (1+\frac{-1\cdot y^2}{2!}+\frac{+1\cdot y^4}{4!}+\cdots )+i(y+\frac{-1\cdot y^3}{3!}+\frac{+1\cdot y^5}{5!}+\cdots )\\ & =& \cos y+i\sin y\\ \therefore e^{ix}& =& \cos x+i\sin x,\text{ Euler's Equation}\end{array}$
  So $theTaylo{r}^{\prime }sEquation,Eule{r}^{\prime }sEquation\text{ are unified in }ComplexSpace$ with $ the\ Trigonometry\ Functions and $NaturalExponentialFunctions$

• Applications of Taylor Series
  The uses of the Taylor series are:

  • Taylor series is used to evaluate the value of a whole function in each point,
    if the functional values and derivatives are identified at a single point.
  • The representation of Taylor series reduces many mathematical proofs.
    The sum of partial series can be used as an approximation of the whole series.
  • Multivariate Taylor series is used in many optimization techniques.
    This series is used in the power flow analysis of electrical power systems.

Maclaurin Series:

• $\text{First define }{f}^{(0)}(x)=f(x)\text{ and }0!=1$ :
• Formula: $\begin{array}{c}\\ f(x)& =& \sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n\\ & =& f(0)+\sum _{n=1}^{\mathrm{\infty }}\frac{f^{(n)}(0)}{n!}{x}^n\\ \end{array}$

幂级数拟合:

幂函数: $x^n,n\in N,x\in R$ 这类函数, 例如 $x^2,x^3,...$
其优点:

• 容易计算及实现:
  计算上, 只要对 x 及系数$(+,-,\times ,÷)$就可快速高性能求值; 甚至 口/心/笔 算可得出结果;
  实现上, 对硬件没特别要求, 嵌入式设备上都可快速实现, 并已有许多可直接使用的成熟软硬件库);
• 任意拟合精度: 预先确定可拟合至任意精度, 并可根据需要设置:
• 幂函数的特点(properties)为人熟知使用:
  例如:
  $y=a\cdot x^2+bx+c$ 的 U形图, 有极值以及求解极值.
  $y=a\cdot x^3+b{x}^2+cx+d$ 的 两个极值以及求解极值 .
  ...

Taylor Series幂级数(Taylor Series)证明

$\text{First define }{f}^{(0)}(x)=f(x)\text{ and }0!=1$ :
$\begin{array}{l}\\ \text{hypothesis: }\\ \text{ 1. the function }f(x)\text{ has }derivativesofeveryorder\text{ and that we can in fact find them all. }\\ \text{ 2. the function }f(x)\text{ does in fact have }apowerseries\text{representation about }x=x_0.\end{array}$

$\begin{array}{r}\\ \text{such that:}& \\ \mathrm{\exists }f(x)& =P(x),\mathrm{\forall }f(x)\\ \text{where}P(x)& =\sum _{n=0}^{\mathrm{\infty }}{c}_n(x-x_0{)}^n\\ & =c_0(x-x_0{)}^0+c_1(x-x_0{)}^1+c_2(x-x_0{)}^2+...+c_n(x-x_0{)}^n+...\\ & =c_0+\sum _{n=1}^{\mathrm{\infty }}{c}_n(x-x_0{)}^n\\ c_0& =f(x_0)\\ P^{(0)}(x)& =f^{(0)}(x)\\ P^{(1)}(x)& =f^{(1)}(x)\\ P^{(2)}(x)& =f^{(2)}(x)\\ P^{(3)}(x)& =f^{(3)}(x)\\ & ...\\ P^{(n)}(x)& =f^{(n)}(x)\end{array}$

$\begin{array}{r}\\ P^{(0)}(x)& =& c_0& +& c_1(x-x_0{)}^1& +& c_2(x-x_0{)}^2& +& c_3(x-x_0{)}^3& +& ...& +& c_n(x-x_0{)}^n& +& ...\\ P^{(1)}(x)& =& 0& +& 1!c_1& +& 2!c_2(x-x_0{)}^1& +& 3c_3(x-x_0{)}^{(3-1)}& +& ...& +& n{c}_n(x-x_0{)}^{(n-1)}& +& ...\\ P^{(2)}(x)& =& 0& +& 0& +& 2!c_2& +& 3!c_3(x-x_0{)}^{(3-2)}& +& ...& +& n(n-1)c_n(x-x_0{)}^{(n-2)}& +& ...\\ P^{(3)}(x)& =& 0& +& 0& +& 0& +& 3!c_3& +& ...& +& n(n-1)(n-2)c_n(x-x_0{)}^{(n-3)}& +& ...\\ & ...& \\ P^{(n)}(x)& =& 0& +& 0& +& 0& +& 0& +& ...& +& n!c_n& +& ...\\ & ...& \\ \text{NOW plugin in }x& =& x_0\text{ , and we have: }& & \\ P^{(0)}(x_0)& =& f^{(0)}(x_0)=0!c_0& & \\ P^{(1)}(x_0)& =& f^{(1)}(x_0)=1!c_1& & \\ P^{(2)}(x_0)& =& f^{(2)}(x_0)=2!c_2& & \\ P^{(3)}(x_0)& =& f^{(3)}(x_0)=3!c_3& & \\ & ...& & & \\ P^{(n)}(x_0)& =& f^{(n)}(x_0)=n!c_n& & \\ & ...& & & \\ \\ c_n& =& \frac{f^{(n)}(x_0)}{n!}\\ \end{array}$

$\begin{array}{l}\\ \text{coefficients}& \\ & where& c_n=\frac{f^{(n)}(x_0)}{n!}\\ \end{array}$

$\text{First define }{f}^{(0)}(x)=f(x)\text{ and }0!=1$ :
$\begin{array}{c}\\ f(x)& =& \sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n\\ \end{array}$

Frequently Asked Questions – FAQs

• Q1 What is a Taylor series?
  Taylor series is a function of an infinite sum of terms in increasing order of degree. Taylor series of polynomial functions is a polynomial.
• Q2 What is the use of Taylor series?
  Taylor series is used to evaluate the value of a whole function in each point if the functional values and derivatives are identified at a single point.
• Q3 What is a Maclaurin series?
  A Taylor series is called Maclaurin series when the function is centered at zero point.