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Ряд Тейлора

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Формула Тейлора

\(f(x)=f(a)+\displaystyle\frac{f'(a)}{1!}(x-a)+\frac{f”(a)}{2!}(x-a)^2+…+\frac{f^{(n)}(a)}{n!}(x-a)^n+o((x-a)^n)\)

 Основные разложения в ряд Тейлора

1. \(\quad e^x=1+\displaystyle\frac{x}{1!}+\frac{x^2}{2!}+…+\frac{x^n}{n!}+…, x\in R\)
2. \(\quad \sin{x}=x-\displaystyle\frac{x^3}{3!}+\frac{x^5}{5!}-…+(-1)^{n-1}\frac{x^{2n-1}}{(2n-1)!}+…, x\in R\)
3. \(\quad \cos{x}=1-\displaystyle\frac{x^2}{2!}+\frac{x^4}{4!}-…+(-1)^n\frac{x^{2n}}{(2n)!}+…,x\in R\)
4. \(\quad \mathrm{ln}(1+x)=x-\displaystyle\frac{x^2}{2}+\frac{x^3}{3}-…+(-1)^{n-1}\frac{x^n}{n}+…,x\in(-1;1]\)
5. \(\quad \mathrm{ln}(1-x)=-x-\displaystyle\frac{x^2}{2}-\frac{x^3}{3}-…-\frac{x^n}{n}-…,x\in[-1;1)\)
6. \(\quad (1+x)^{\alpha}=1+\alpha x+\displaystyle\frac{\alpha(\alpha-1)}{2}x^2+…+\frac{\alpha(\alpha-1)…(\alpha-n+1)}{n!}x^n+…,x\in(-1;1)\)
7. \(\quad \mathrm{arctg}{x}=x-\displaystyle\frac{x^3}{3}+\frac{x^5}{5}-…+(-1)^{n-1}\frac{x^{2n-1}}{2n-1}+…,x\in[-1;1]\)
8. \(\quad \arcsin{x}=x+\displaystyle\frac{1}{2\cdot3}x^3+\frac{1\cdot3}{2\cdot4\cdot5}x^5+…+\frac{(2n-1)!!}{(2n)!!}\frac{x^{2n+1}}{2n+1}+…,x\in[-1;1]\)
9. \(\quad \mathrm{sh}{x}=x+\displaystyle\frac{x^3}{3!}+\frac{x^5}{5!}+…+\frac{x^{2n-1}}{(2n-1)!}+…,x\in R\)
10. \(\quad \mathrm{ch}{x}=x+\displaystyle\frac{x^2}{2!}+\frac{x^4}{4!}+…+\frac{x^{2n}}{(2n)!}+…,x\in R\)
11. \(\quad \displaystyle\frac{1}{1+x}=1-x+x^2-…+(-1)^nx^n+…,x\in (-1;1)\)
12. \(\quad \displaystyle\frac{1}{1-x}=1+x+x^2+…+x^n+…,x\in (-1;1)\)
13. \(\quad \sqrt{1+x}=1+\displaystyle\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3-…+(-1)^{n-1}\frac{(2n-3)!!}{(2n)!!}x^n+…,x\in [-1;1]\)
14. \(\quad \displaystyle\frac{1}{(1-x)^2}=1+2x+3x^2+…+(n+1)x^n+…,x\in (-1;1)\)