Формула Тейлора 28.11.2025

Вывод коэффициентов формулы Тейлора

$$\begin{array}{c}f(x)\in D^n(a)\\ f(x)=C_0+C_1(x-a)+C_2(x-a{)}^2+\cdots +C_n(x-a{)}^n+r_n(x)\\ f(x)=P_n(x)+r_n(x),r_n(x)\to 0\text{ при }x\to a-\text{остаток}\\ f(a)=P(a)\\ (\ast )f^{\prime }(a)=P^{\prime }(a)\dots f^{(n-1)}(a)=P^{(n-1)}(a),f^{(n)}=P^{(n)}(a)\\ f(a)=C_0+r_n(a)=C_0\\ \text{Заметим, что }{r}_n(x)\in C(a)\text{, так как }{r}_n=f(x)-P_n(x)\in D(a)\\ \Rightarrow r_n(a)=0\\ \\ \text{По }(\ast )\Rightarrow f^{\prime }(a)=P^{\prime }(a),f^{(n)}(a)=P^{(n)}(a)\\ P^{\prime }(x)=C_1+C_2\cdot 2(x-a)+C_3\cdot 3(x-a{)}^2+\cdots +C_n\cdot n(x-a{)}^{n-1}\\ P^{\prime }(a)=C_1\Rightarrow C_1=f^{\prime }(a)\\ \\ \text{По }(\ast )\Rightarrow f^{\prime \prime }(a)=P^{\prime \prime }(a),f^{(n)}(a)=P^{(n)}(a)\\ P^{\prime \prime }(x)=2C_2+C_3\cdot 3\cdot 2(x-a)+C_4\cdot 4\cdot 3(x-a{)}^2+\cdots +C_n\cdot n(n-1)(x-a{)}^{n-1}\\ P^{\prime \prime }(a)=2C_2\Rightarrow C_2=\frac{f^{\prime \prime }(a)}{2}\\ \\ \text{По }(\ast )\Rightarrow f^{\prime \prime \prime }(a)=P^{\prime \prime \prime }(a),f^{(n)}(a)=P^{(n)}(a)\\ P^{\prime \prime \prime }(x)=C_3\cdot 3\cdot 2+C_4\cdot 4\cdot 3\cdot 2(x-a)+\cdots +C_n\cdot n(n-1)(n-2)(x-a{)}^{n-3}\\ P^{\prime \prime \prime }(a)=2\cdot 3\cdot C_3\Rightarrow C_3=\frac{f^{\prime \prime \prime }(a)}{3!};C_4=\frac{f^{IV}(a)}{4!};C_5=\frac{f^V(a)}{5!};\dots ;C_n=\frac{f^{(n)}(a)}{n!};\\ \\ \text{в }U(a)\\ f(x)=P_n(x)+r_n(x)\\ r_n(x)=o((x-a{)}^n)?\\ r_n(x)=f(x)-P_n(x)\in D^n(a)\\ \underset{x\to a}{\lim }\frac{r_n(x)}{(x-a{)}^n}=\left[\frac{0}{0}\right]\stackrel{L^{\prime }H}{=}\underset{x\to a}{\lim }\frac{r_n^{\prime }(x)}{n(x-a{)}^{n-1}}=\cdots =\underset{x\to a}{\lim }\frac{r_n^{(n)}(x)}{n!}=\\ =\underset{x\to a}{\lim }\frac{f^{(n)}(x)-P_n^{(n)}(x)}{n!}=0\\ \Rightarrow r_n(x)=\overline{\stackrel{ˉ}{o}}((x-a{)}^n)\end{array}$$

Многочлен Тейлора степени n

Многочлен, соответствующий функции $f(x)$:

$$P_n(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\cdots +\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$

Формула Тейлора

$$f(x)=\sum _{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k+\overline{\stackrel{ˉ}{o}}((x-a{)}^n)$$

$r_n(x)$ — остаточный член ($0!=1$)

1. Форма Пеано: $r_n(x)=\overline{\stackrel{ˉ}{o}}((x-a{)}^n)$

2. Форма Лагранжа: $r_n(x)=\frac{1}{(n+1)!}{f}^{(n+1)}(a+\theta (x-a))\cdot (x-a{)}^{n+1},0<\theta <1$

Свойства $\overline{\stackrel{ˉ}{o}}$

1. $\overline{\stackrel{ˉ}{o}}(f)\pm \overline{\stackrel{ˉ}{o}}(f)=\overline{\stackrel{ˉ}{o}}(f)$
2. $\overline{\stackrel{ˉ}{o}}(c\cdot f)=\overline{\stackrel{ˉ}{o}}(f),c\ne 0$
3. $c\cdot \overline{\stackrel{ˉ}{o}}(f)=\overline{\stackrel{ˉ}{o}}(f),c\ne 0$
4. $\overline{\stackrel{ˉ}{o}}(\overline{\stackrel{ˉ}{o}}(f))=\overline{\stackrel{ˉ}{o}}(f)$

Примеры:

• $\overline{\stackrel{ˉ}{o}}(x^3)+\overline{\stackrel{ˉ}{o}}(x^2)=\overline{\stackrel{ˉ}{o}}(x^2)$
• $\overline{\stackrel{ˉ}{o}}(x^3)+100500\overline{\stackrel{ˉ}{o}}(x^2)=\overline{\stackrel{ˉ}{o}}(x^2)$
• $x\cdot \overline{\stackrel{ˉ}{o}}(x^2)=\overline{\stackrel{ˉ}{o}}(x^3)$
• $\overline{\stackrel{ˉ}{o}}(x^2)\cdot \overline{\stackrel{ˉ}{o}}(x^3)=\overline{\stackrel{ˉ}{o}}(x^5)$

Формула Маклорена

Формула Тейлора с $a=0$ в окрестности $U(0)$:

$$f(x)=\sum _{k=0}^n\frac{f^{(k)}(0)}{k!}{x}^k+r_n(x)$$

Стандартные разложения

1. Экспонента:

$$e^x=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}+\overline{\stackrel{ˉ}{o}}(x^n)=\sum _{k=0}^n\frac{f^{(k)}(0)}{k!}{x}^k+\overline{\stackrel{ˉ}{o}}(x^n)$$

2. Косинус:

$$\cos x=1-\frac{x^2}{2}+\frac{x^4}{4!}+\cdots +(-1{)}^n\frac{x^{2n}}{(2n)!}+\overline{\stackrel{ˉ}{o}}(x^{2n+1})$$

3. Синус:

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots +(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}+\overline{\stackrel{ˉ}{o}}(x^{2n+2})=\sum _{k=0}^n(-1{)}^k\frac{x^{2k+1}}{(2k+1)!}+\overline{\stackrel{ˉ}{o}}(x^{2n+2})$$

4. Степенная функция:

$$(1+x{)}^{\mu }=1+\mu x+\frac{\mu (\mu -1)}{2}{x}^2+\cdots +\frac{\mu (\mu -1)\dots (\mu -n+1)}{n!}{x}^n+\overline{\stackrel{ˉ}{o}}(x^n)$$

5. Дробь (плюс):

$$\frac{1}{1+x}=1-x+x^2-x^3+\cdots +(-1{)}^n{x}^n+\overline{\stackrel{ˉ}{o}}(x^n)$$

6. Дробь (минус):

$$\frac{1}{1-x}=1+x+x^2+x^3+\cdots +x^n+\overline{\stackrel{ˉ}{o}}(x^n)$$

7. Натуральный логарифм:

$$\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots +\frac{(-1{)}^{n-1}\cdot x^n}{n}+\overline{\stackrel{ˉ}{o}}(x^n)$$