Power Series: Complete Guide

Understanding Infinite Polynomials and Their Applications

What is a Power Series?

A power series is an infinite series of the form that represents a function as an "infinite polynomial" centered around a specific point.

f(x) = a₀ + a₁(x-c) + a₂(x-c)² + a₃(x-c)³ + ... = ∑ aₙ(x-c)ⁿ

Key Components

Coefficients (aₙ)

Constants that determine the shape and behavior of the series

Center (c)

The point around which the series is expanded

Powers (x-c)ⁿ

The polynomial terms that make up the series

Index (n)

The term position, starting from n=0 to infinity

Fundamental Examples

1

Geometric Series

The most fundamental power series:

1/(1-x) = 1 + x + x² + x³ + ... = ∑ xⁿ

This converges when |x| < 1 and diverges when |x| ≥ 1

2

Exponential Function

The exponential function has a beautiful power series expansion:

eˣ = 1 + x + x²/2! + x³/3! + ... = ∑ xⁿ/n!

This series converges for all real numbers x

3

Sine and Cosine

Trigonometric functions also have power series representations:

sin(x) = x - x³/3! + x⁵/5! - ... = ∑ (-1)ⁿ x²ⁿ⁺¹/(2n+1)!

cos(x) = 1 - x²/2! + x⁴/4! - ... = ∑ (-1)ⁿ x²ⁿ/(2n)!

Convergence and Radius of Convergence

A power series doesn't necessarily converge for all values of x. The set of x-values for which the series converges is called the interval of convergence.

Radius of Convergence (R)

For a power series centered at c, there exists a number R (possibly 0 or ∞) called the radius of convergence such that:

|x-c| < R

The series converges absolutely

|x-c| > R

The series diverges

|x-c| = R

Convergence must be checked case by case

Finding the Radius of Convergence

Using the Ratio Test:

R = lim |aₙ/aₙ₊₁| as n→∞

Or using the Root Test:

R = 1/lim sup |aₙ|¹/ⁿ as n→∞

Important Power Series Expansions

Function Power Series Expansion Interval of Convergence eˣ ∑ xⁿ/n! = 1 + x + x²/2! + x³/3! + ... All real numbers (-∞, ∞) sin(x) ∑ (-1)ⁿ x²ⁿ⁺¹/(2n+1)! = x - x³/3! + x⁵/5! - ... All real numbers (-∞, ∞) cos(x) ∑ (-1)ⁿ x²ⁿ/(2n)! = 1 - x²/2! + x⁴/4! - ... All real numbers (-∞, ∞) 1/(1-x) ∑ xⁿ = 1 + x + x² + x³ + ... |x| < 1 ln(1+x) ∑ (-1)ⁿ⁺¹ xⁿ/n = x - x²/2 + x³/3 - x⁴/4 + ... -1 < x ≤ 1 arctan(x) ∑ (-1)ⁿ x²ⁿ⁺¹/(2n+1) = x - x³/3 + x⁵/5 - ... |x| ≤ 1

Operations on Power Series

+

Addition and Subtraction

Power series can be added term by term within their common interval of convergence:

∑ aₙxⁿ ± ∑ bₙxⁿ = ∑ (aₙ ± bₙ)xⁿ

×

Multiplication

Power series can be multiplied using the Cauchy product:

(∑ aₙxⁿ)(∑ bₙxⁿ) = ∑ cₙxⁿ

where cₙ = a₀bₙ + a₁bₙ₋₁ + ... + aₙb₀

d/dx

Differentiation

Power series can be differentiated term by term within their radius of convergence:

d/dx [∑ aₙxⁿ] = ∑ n·aₙxⁿ⁻¹

∫

Integration

Power series can be integrated term by term within their radius of convergence:

∫ [∑ aₙxⁿ] dx = C + ∑ aₙxⁿ⁺¹/(n+1)

Applications of Power Series

Approximating Functions

Power series provide polynomial approximations of complicated functions. The more terms used, the better the approximation.

Solving Differential Equations

Many differential equations can be solved by assuming a power series solution and determining the coefficients.

Numerical Analysis

Power series enable efficient computation of function values, derivatives, and integrals in computer algorithms.

Physics and Engineering

Used in quantum mechanics, electromagnetism, and signal processing to model complex physical phenomena.

Probability and Statistics

Moment generating functions and characteristic functions use power series to represent probability distributions.

Computer Science

Generating functions use power series to solve counting problems and analyze algorithms.

Taylor and Maclaurin Series

A Taylor series is a specific type of power series where the coefficients are determined by the function's derivatives at the center point.

Taylor Series Formula

f(x) = f(c) + f'(c)(x-c) + f''(c)(x-c)²/2! + f'''(c)(x-c)³/3! + ... = ∑ f⁽ⁿ⁾(c)(x-c)ⁿ/n!

Maclaurin Series (Special Case)

When the center c = 0, we get a Maclaurin series:

f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... = ∑ f⁽ⁿ⁾(0)xⁿ/n!

Taylor's Theorem

Taylor's theorem provides the remainder term, telling us how accurate our approximation is:

Rₙ(x) = f⁽ⁿ⁺¹⁾(z)(x-c)ⁿ⁺¹/(n+1)!

where z is some number between x and c.

Key Properties and Theorems

Uniqueness

If two power series converge to the same function on an interval, they must have identical coefficients.

Analytic Functions

A function is called analytic at a point if it can be represented by a power series in some neighborhood of that point.

Identity Theorem

If an analytic function has a power series with all zero coefficients, then the function is identically zero.

Why Power Series are Powerful

Power series transform complicated functions into infinite polynomials, allowing us to:

• Approximate functions with polynomials • Differentiate and integrate term by term • Solve otherwise intractable equations • Understand function behavior near points