Common Taylor Polynomials

Function:

$$f\left(x\right)=e^{x}$$

Taylor Polynomial:

$$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots+\frac{x^{n}}{n!}$$

Sigma Notation:

$$\sum_{k=0}^{n}\frac{x^{k}}{k!}$$

Center:

0

Function:

$$f\left(x\right)=\ln x$$

Taylor Polynomial:

$$\left(x-1\right)-\frac{\left(x-1\right)^2}{2}+\frac{\left(x-3\right)^3}{3}-\frac{\left(x-1\right)^4}{4}+\cdots+\frac{\left(-1\right)^{n+1}\left(x-1\right)^{n}}{n}$$

Sigma Notation:

$$\sum_{k=0}^{n}\frac{\left(-1\right)^{k+1}\left(x-1\right)^{k}}{k}$$

Center:

1

Function:

$$f\left(x\right)=\sin x$$

Taylor Polynomial:

$$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots+\frac{\left(-1\right)^{n}x^{2n+1}}{\left(2n+1\right)!}$$

Sigma Notation:

$$\sum_{k=0}^{n}\frac{\left(-1\right)^{k}x^{2k+1}}{\left(2k+1\right)!}$$

Center:

0

Function:

$$f\left(x\right)=\cos x$$

Taylor Polynomial:

$$1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots+\frac{\left(-1\right)^{n}x^{2n}}{\left(2n\right)!}$$

Sigma Notation:

$$\sum_{k=0}^{n}\frac{\left(-1\right)^{k}x^{2k}}{\left(2k\right)!}$$

Center:

0

Function:

$$f\left(x\right)=\frac{1}{1-x}$$

Taylor Polynomial:

$$1+x+x^2+x^3+\cdots+x^{n}$$

Sigma Notation:

$$\sum_{k=0}^{n}x^{k}$$

Center:

0