Power Series

Definition

A power series is an infinite series of the form $\sum_{n=0}^{\infty} c_n (x-a)^n$, which can be viewed as an “infinite polynomial” centered at $a$. It defines a function $f(x)$ for all $x$ within its interval of convergence.

• How to read: “The infinite sum from n equals zero to infinity of the coefficient c n times the quantity x minus a raised to the nth power.”
• Meaning: An infinite polynomial centered at $a$ that represents $f(x)$ inside its radius of convergence $R$.

Why It Matters

We can’t solve most real-world math exactly. Power series are the “Cheat Code” that lets us turn any “hard” function into a simple “infinite polynomial.” If your computer didn’t use this math, it couldn’t tell you the sine of an angle or the path of a particle. It is the “Numerical DNA” of modern digital computation.

Core Concepts

• Radius of Convergence ($R$): A value such that the series converges for $|x-a| < R$ and diverges for $|x-a| > R$. It is typically found using the Ratio Test.
  • How to read: “The absolute value of the difference between x and a is strictly less than the radius of convergence R; or the absolute value of the difference between x and a is strictly greater than the radius of convergence R.”
  • Meaning: $R$ is the distance from center $a$ to the nearest singularity; endpoints $a \pm R$ must be checked separately.
• Interval of Convergence: The set of all $x$ for which the series converges, including potential endpoints $a \pm R$.
• Term-by-Term Operations:
  • Differentiation: $f'(x) = \sum n c_n (x-a)^{n-1}$.
  • Integration: $\int f(x) dx = \sum \frac{c_n}{n+1} (x-a)^{n+1} + C$.
  • How to read: “The derivative of the function f with respect to x is equal to the sum of n times the coefficient c n times the quantity x minus a raised to the power of n minus one; and the indefinite integral of the function f of x with respect to x is equal to the sum of the coefficient c n divided by the quantity n plus one, times the quantity x minus a raised to the power of n plus one, plus a constant of integration C.”
  • Meaning: Inside the interval of convergence, power series behave like polynomials — enabling series solutions of ODEs and term-by-term integration of non-elementary integrands.
• Both differentiation and integration preserve the radius of convergence $R$.

Connected Concepts

• Complex Power Series
• Power-Series Solutions
• Geometric Sequence and Geometric Series