Taylor Series

Definition

A Taylor Series is a representation of a smooth function $f(x)$ as an infinite sum of terms calculated from the values of the function’s derivatives at a single center point $x = a$: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$

• How to read: “The function f of x equals the sum from n equals zero to infinity of the n-th derivative of f evaluated at a, divided by n factorial, all times the quantity x minus a raised to the n-th power.”
• Meaning / when to use: Use to approximate a differentiable function locally near a point $a$ using a polynomial. The $n!$ factor in the denominator accounts for the derivatives of the power terms during repeated differentiation.

Why It Matters

Taylor series allow complex transcendental functions (like $\sin x$, $\ln x$, or $e^x$) to be approximated as simple polynomial equations. This is how calculators and computers evaluate these functions, how physicists solve non-linear differential equations (by linearizing them), and how numerical integration algorithms approximate integrals that cannot be solved analytically.

Core Concepts

• Derivative Coefficients: The coefficients of the series are determined by the derivatives of the function at the center $a$. The higher the degree of the polynomial, the better the local approximation.
• Center of Expansion ($a$): The specific point about which the function is approximated. The approximation is highly accurate near $a$, but may degrade as $x$ moves further away.
• Convergence Interval: The range of $x$ values for which the infinite series converges to the actual value of the function.
• Taylor Polynomial: A truncated, finite version of the series (e.g., $P_n(x)$) used for practical calculations.

Connected Concepts

• Maclaurin Series
• Taylor’s Formula for Two Variables
• Fundamental Theorem of calculus