Taylor Polynomial

$$-30$$  

$$-15$$  

$$-5$$  

$$-\frac{5}{6}$$  

$$-\frac{1}{6}$$  

$$2-\left(x-3\right)+3\left(x-3\right)^2+2\left(x-3\right)^3$$  

$$2-\left(x-3\right)+3\left(x-3\right)^2+4\left(x-3\right)^3$$  

$$2-\left(x-3\right)+6\left(x-3\right)^2+12\left(x-3\right)^3$$  

$$2-x+3x^2+2x^3$$  

$$2-x+6x^2+12x^3$$  

$$-x-\frac{x^2}{2}-\frac{x^3}{3}$$  

$$1-x+\frac{x^2}{2}$$  

$$x-\frac{x^2}{2}+\frac{x^3}{3}$$  

$$-1+x-\frac{x^2}{2}$$  

$$-x+\frac{x^2}{2}-\frac{x^3}{3}$$  

$$1+\frac{x}{2}$$  

$$1+x$$  

$$1-\frac{x^2}{2}$$  

$$1-2x^2$$  

$$1-2x+x^2$$  

$$\frac{1}{6}$$  

$$\frac{1}{3}$$  

$$1$$  

$$3$$  

$$6$$  

$$-\frac{1}{6}$$  

$$0$$  

$$\frac{1}{120}$$  

$$\frac{1}{6}$$  

$$1$$  

$$x-\frac{x^3}{2!}+\frac{x^5}{4!}-\frac{x^6}{6!}+...$$  

$$x-\frac{4x^3}{2!}+\frac{16x^5}{4!}-\frac{64x^7}{6!}+...$$  

$$2x-\frac{8x^3}{3!}+\frac{32x^5}{5!}-\frac{128x^7}{7!}+...$$  

$$2x^2-\frac{2x^4}{3!}+\frac{2x^6}{5!}-\frac{2x^8}{7!}+...$$  

$$2x^2-\frac{8x^4}{3!}+\frac{32x^6}{5!}-\frac{128x^8}{7!}+...$$  

$$1=f^n\left(c\right);\ 2=n!;\ 3=\left(x-c\right)$$  

$$1=f^n\left(c\right);\ 2=n!;\ 3=x$$  

$$1=f^n\left(0\right);\ 2=n!;\ 3=\left(x-c\right)$$  

$$1=n!;\ 2=f^n\left(c\right);\ 3=\left(x-c\right)$$  

$$\left(21\right)27!$$  

$$\frac{21}{27!}$$  

$$21$$  

$$\left(-21\right)27!$$  

T₈ (1) = 2.71825396825

T₈ (1) = 2.71826407935

T₂₀ (1) = 2.71828182845

T₂₀ (1) = 2.71839293845