Power Series — Mathematics Knowledge System

Overview

A power series is an infinite series whose terms are powers of (x − a) multiplied by coefficients: Σcₙ(x−a)ⁿ. Unlike a numerical series, a power series defines a function of x, and it converges for x within a specific radius of a and diverges outside it. Power series are the algebraic backbone of Taylor series and let you manipulate transcendental functions (differentiate, integrate, solve differential equations) using term-by-term polynomial-like rules.

Intuition

Think of a power series as an 'infinite-degree polynomial' whose coefficients cₙ encode a function. Close to the center a, the powers (x−a)ⁿ shrink rapidly (if |x−a| is small) and the series behaves well; far from a, the powers grow and can overwhelm the coefficients, causing divergence. The radius of convergence R marks exactly where this tug-of-war between shrinking powers and coefficient size tips over — the series converges for |x−a| < R and diverges for |x−a| > R.

Interactive Graph

Formal Definition

Definition

A power series centered at a:

\sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots

cₙ are constant coefficients; the series is a function of x

R = \frac{1}{\lim_{n\to\infty} |c_{n+1}/c_n|}

Radius of convergence via the ratio test (when this limit exists)

Notation

Notation	Meaning
$c_n$	The n-th coefficient of the power series
$R$	Radius of convergence — the series converges for \|x−a\| < R
$(a-R,\ a+R)$	The set of x for which the series converges; endpoints must be checked individually

Derivation

Applying the ratio test directly to a power series Σcₙ(x−a)ⁿ to find its radius of convergence:

\lim_{n\to\infty}\left|\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}\right| = |x-a|\lim_{n\to\infty}\left|\frac{c_{n+1}}{c_n}\right|

Factor out |x-a| from the ratio test limit

\text{Converges when } |x-a| \cdot \lim\left|\frac{c_{n+1}}{c_n}\right| < 1 \iff |x-a| < \frac{1}{\lim|c_{n+1}/c_n|} = R

Solving the ratio-test inequality for x gives exactly the radius of convergence

Properties

Term-by-term differentiation

\frac{d}{dx}\sum_{n=0}^{\infty} c_n(x-a)^n = \sum_{n=1}^{\infty} n c_n (x-a)^{n-1}

Condition: valid within the same radius of convergence R

Term-by-term integration

\int \sum_{n=0}^{\infty} c_n(x-a)^n\,dx = C + \sum_{n=0}^{\infty} \frac{c_n}{n+1}(x-a)^{n+1}

Condition: valid within the same radius of convergence R

Uniqueness

\text{If } \sum c_n(x-a)^n = \sum d_n(x-a)^n \text{ on an interval around } a, \text{ then } c_n = d_n \text{ for all } n

Applications

Solving linear differential equations (e.g. the quantum harmonic oscillator, Legendre's equation) via the power series (Frobenius) method, assuming a solution y = Σcₙxⁿ and matching coefficients.

Worked Examples

Apply the ratio test to |aₙ₊₁/aₙ|.
$\left|\frac{(x-2)^{n+1}/(n+1)^2}{(x-2)^n/n^2}\right| = |x-2|\cdot\frac{n^2}{(n+1)^2} \to |x-2|$
Converges when |x-2| < 1, so R = 1, centered at a=2.
$R = 1, \quad 1 < x < 3 \text{ (before endpoint check)}$
Check endpoints: at x=1 and x=3, series becomes Σ(±1)ⁿ/n², which converges absolutely (p-series, p=2>1) at both.
$\text{Interval of convergence: } [1,3]$

Answer: R = 1, interval of convergence [1, 3]

Practice Problems

Difficulty 5/10

Find the radius of convergence of Σ xⁿ/n!.

Difficulty 6/10

Find the interval of convergence of Σ (-1)ⁿ (x+1)ⁿ / n.

Common Mistakes

Common Mistake

Forgetting to check the endpoints of the interval of convergence separately.

The ratio/root test only determines the open interval (a-R, a+R). Each endpoint x=a±R must be substituted in and tested individually with a different convergence test (usually p-series or alternating series test), since the ratio test itself is inconclusive there (L=1).

Common Mistake

Assuming term-by-term differentiation or integration changes the radius of convergence.

Differentiating or integrating a power series term-by-term preserves the SAME radius of convergence R, though convergence AT the endpoints can change.

Quiz

The radius of convergence R of a power series Σcₙ(x−a)ⁿ tells you:

the value the series sums tothe distance from the center a within which the series convergesthe number of terms neededthe derivative of the sum

Why are power series so useful in computation (calculators, simulations)?

They make functions longerThey approximate complicated functions (eˣ, sin x, etc.) by polynomials that computers evaluate with +, −, ×They only work for polynomialsThey avoid all error

Historical Background

Power series appeared implicitly in Newton's and Gregory's work on infinite series expansions in the 1660s-70s, well before the underlying convergence theory existed. The rigorous notion of a radius of convergence was developed by Cauchy in the 1820s and sharpened by Abel, whose 1826 theorem on the behavior of power series at the boundary of convergence remains a cornerstone result.

1670s
Newton and Gregory manipulate infinite series expansions of functions
Isaac Newton, James Gregory
1821
Cauchy formalizes convergence and introduces the radius of convergence concept
Augustin-Louis Cauchy
1826
Abel proves his theorem on continuity of power series at the boundary of convergence
Niels Henrik Abel

Summary

A power series Σcₙ(x−a)ⁿ defines a function of x, converging within a radius R of the center a.
The ratio test on the coefficients typically determines R; endpoints x = a ± R must be checked separately.
Within the radius of convergence, power series can be differentiated and integrated term by term, preserving R.
Power series coefficients are unique — two power series equal on an interval must have identical coefficients.
Power series underlie Taylor series and the power-series (Frobenius) method for solving differential equations.

References

WebsiteWikipedia — Power series