Taylor Series Polynomials

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

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

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

$$2-x+3x^2+2x^3$$  

$$2-x+6x^2+12x^3$$  

$$-\frac{1}{6}$$  

$$0$$  

$$\frac{1}{120}$$  

$$\frac{1}{6}$$  

$$1$$  

-2/3

-4/3

4/3

-8/3

2/3

Converges to a sum of 10/7.

Converges, but sum cannot be determined.

Diverges.

$$P_4\left(x\right)=x-\frac{1}{6}x^3+\frac{1}{5!}x^5-\frac{1}{7!}x^7$$  

$$P_4\left(x\right)=-x^2+\frac{1}{6}x^4$$  

$$P_4\left(x\right)=x^2-\frac{1}{6}x^4$$  

$$P_4\left(x\right)=2x^2-4x^4$$