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Tony's Intertwingled Memex • Reference Manual

Power Series

Formulas & Definitions
Core Definition

An infinite series of the form:

∑n=0∞cn(x−a)n=c0+c1(x−a)+c2(x−a)2+…\sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \dots
where xx is a variable, cnc_n are coefficients, and aa is a constant center. A special type is the Taylor Series representation of a smooth function f(x)f(x) centered at aa:
f(x)=∑n=0∞f(n)(a)n!(x−a)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n

Concept Explanation

Power series approximate complex transcendental functions (like ex,sin⁡x,ln⁡xe^x, \sin x, \ln x) using infinite polynomial combinations. The Radius of Convergence (RR) determines the range over which the series accurately matches the target function.

Visual / Geometric Intuition

Interactive Desmos Graph

Engineering Applications

Engineers use power series to solve differential equations in mechanical vibration, construct digital signal filters, and approximate structural stress calculations when exact analytic formulas do not exist.

Example Problem

Find the Taylor series (Maclaurin series) for f(x)=exf(x) = e^x centered at a=0a = 0.

Solution

Step 1: Compute derivatives of f(x)f(x)

f(n)(x)=exfor all nf^{(n)}(x) = e^x \quad \text{for all } n

Step 2: Evaluate derivatives at center a=0a = 0

f(n)(0)=e0=1for all nf^{(n)}(0) = e^0 = 1 \quad \text{for all } n

Step 3: Construct the Taylor series

f(x)=∑n=0∞f(n)(0)n!xn=∑n=0∞xnn!f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = \sum_{n=0}^{\infty} \frac{x^n}{n!}
ex=1+x+x22!+x33!+…e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots

Connections & References