Power Series

where $x$ is a variable, $c_n$ are coefficients, and $a$ is a constant center. A special type is the Taylor Series representation of a smooth function $f(x)$ centered at $a$:

Concept Explanation

Power series approximate complex transcendental functions (like $e^x, \sin x, \ln x$) using infinite polynomial combinations. The Radius of Convergence ($R$) 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 $f(x) = e^x$ centered at $a = 0$.

Solution

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

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

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

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

Step 3: Construct the Taylor series

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = \sum_{n=0}^{\infty} \frac{x^n}{n!}$$

$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$

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Connections & References

• Parent Concepts: Improper Integrals
• Sub-concepts: Vectors
• Course Links: Calc-2 Session 19 | Calc-2 Session 22