Taylor Series

A method of expressing a function as a Power Series of that function’s derivatives. Allows for approximating a function if the derivatives are known.

Formal Definition

• For a given function $f(x)$
• Considering power series ${\sum }_{n=1}^{\infty }{C}_n(x-a{)}^n$
• Note $C_n=\frac{f^n(a)}{n!}$ for any $n$ ($f^n$ being the n-th derivative)
• Then, $f(x)={\sum }_{n=1}^{\infty }\frac{f^n(a)}{n!}(x-a{)}^n$ is a Taylor series for $f$ at $a$

Intuition

Concepts

• Maclaurin Series
• Taylor’s Theorem in One Dimension
• Taylor’s Theorem in One Dimension Constant Version
• First-Order Multivariable Taylor Series

Taylor Approximations List

• Taylor Approximation for Exponential with Euler’s Base
• Taylor Approximation for Sin
• Taylor Approximation for Cos
• Taylor Approximation for Arctan
• Taylor Approximation for Inverse 1 minus X
• Taylor Approximation for Inverse 1 plus X
• Taylor Approximation for Natural Logarithm 1 plus X