Power Series and Radius of Convergence

⚡ Key Takeaways

Power series: an infinite series $\sum_{n=0}^\infty a_n (x-c)^n$ that behaves like an "infinite polynomial." Every such series has a radius of convergence $R \in [0,\infty]$ such that it converges absolutely for $|x-c| < R$ and diverges for $|x-c| > R$ . The number $R$ is computed by the Cauchy–Hadamard formula. Power series are the foundation of analytic function theory, Taylor expansions, generating functions, and the solution of differential equations. Central results due to Cauchy, Abel, Hadamard, and Weierstrass (19th century).

💡 The Core Insight

The convergence of a power series is governed by a single threshold $R$ . Inside the disk $|x-c|<R$ the series converges absolutely and the geometric "tail control" $|a_n(x-c)^n| \sim (\rho/R)^n$ for $\rho<R$ makes everything well-behaved (continuity, differentiability, integrability term-by-term). Outside the terms fail to even approach zero, so divergence is automatic. The boundary $|x-c|=R$ is the genuinely subtle frontier where anything can happen. The magic is that $R$ depends only on the growth rate of $|a_n|^{1/n}$ , not on $x$ .

DefinitionPower Series

A power series centered at $c \in \mathbb{R}$ (or $\mathbb{C}$ ) with coefficients $a_n$ is the formal sum

1

\quad f(x) = \sum_{n=0}^{\infty} a_n (x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots

Derivation hint: By definition; the symbols are $a_n$ (the $n$ -th coefficient, a fixed scalar), $c$ (the center), $x$ (the variable), $n$ (the summation index running over non-negative integers). The convergence set is $S = \{x : \text{the series } [1] \text{ converges}\}$ . We always have $c \in S$ since at $x=c$ the series reduces to $a_0$ .

DefinitionRadius and Interval of Convergence

The radius of convergence is

2

\quad R = \sup\{\, r \ge 0 : \sum_{n=0}^\infty a_n (x-c)^n \text{ converges for some } |x-c|=r \,\}

with the conventions $R=0$ (converges only at $c$ ) and $R=\infty$ (converges everywhere). The interval of convergence is the set $S$ itself, which is one of $(c-R,c+R)$ , $[c-R,c+R)$ , $(c-R,c+R]$ , or $[c-R,c+R]$ over $\mathbb{R}$ — equal to the open interval together with whichever endpoints converge. Derivation hint: $R$ is defined as a supremum over radii of convergence; the four-fold interval cases arise because endpoint behavior is independent on each side.

TheoremCauchy–Hadamard Formula

For any power series $[1]$ , the radius of convergence is given by

3

\quad \frac{1}{R} = \limsup_{n\to\infty} |a_n|^{1/n}

with the conventions $1/0 = \infty$ and $1/\infty = 0$ . Moreover the series converges absolutely for $|x-c|<R$ and diverges for $|x-c|>R$ . Derivation hint: Apply the root test to $\sum |a_n(x-c)^n|$ ; the $n$ -th root of the general term is $|a_n|^{1/n}|x-c|$ , whose limsup is $|x-c|/R$ .

TheoremRatio Form

If the limit exists, the radius of convergence may also be computed as

4

\quad R = \lim_{n\to\infty} \left| \frac{a_n}{a_{n+1}} \right|

provided the $a_n$ are eventually nonzero and the limit (in $[0,\infty]$ ) exists. This always agrees with $[3]$ when the ratio limit exists, but the root formula $[3]$ is universally valid. Derivation hint: Apply the ratio test to consecutive terms $a_{n+1}(x-c)^{n+1} / a_n(x-c)^n = (a_{n+1}/a_n)(x-c)$ ; convergence requires $|x-c| < \lim|a_n/a_{n+1}|$ .

ProofCauchy–Hadamard via the Root Test

We prove $[3]$ . Let $L = \limsup_{n\to\infty} |a_n|^{1/n}$ and set $R = 1/L$ .

Step 1 — Set up the root test. Apply the root test to the series of absolute values $\sum_{n=0}^\infty |a_n(x-c)^n| = \sum_n |a_n|\,|x-c|^n$ . The relevant quantity is

5

\quad \alpha := \limsup_{n\to\infty} \big( |a_n|\,|x-c|^n \big)^{1/n} = |x-c| \cdot \limsup_{n\to\infty} |a_n|^{1/n} = |x-c|\,L = \frac{|x-c|}{R}.

Derivation hint: $|x-c|$ is a constant factor that pulls out of the limsup of an $n$ -th root; the remaining factor is the definition of $L$ .

Step 2 — Convergence when $|x-c|<R$ . Then $\alpha = |x-c|/R < 1$ . The root test states that if $\limsup (\text{term})^{1/n} < 1$ the series converges absolutely. Concretely, pick $\beta$ with $\alpha < \beta < 1$ ; for all large $n$ we have $|a_n||x-c|^n \le \beta^n$ , and $\sum \beta^n$ converges as a geometric series. By comparison, $[1]$ converges absolutely.

Step 3 — Divergence when $|x-c|>R$ . Then $\alpha > 1$ . By definition of $\limsup$ , infinitely many terms satisfy $(|a_n||x-c|^n)^{1/n} > 1$ , i.e. $|a_n(x-c)^n| > 1$ . The general term does not approach $0$ , so the series diverges (the $n$ -th term test).

Step 4 — Boundary undetermined. When $|x-c| = R$ , $\alpha = 1$ and the root test is inconclusive; both convergence and divergence are possible (see the examples). This establishes $[2]$ and $[3]$ . $\blacksquare$

∎

ProofTerm-by-Term Differentiation Inside R

A crucial consequence: inside the disk of convergence, $f$ is differentiable and

6

\quad f'(x) = \sum_{n=1}^\infty n\, a_n (x-c)^{n-1}, \qquad |x-c| < R,

and the differentiated series has the same radius $R$ . Derivation hint: $\limsup |n a_n|^{1/n} = \limsup n^{1/n}|a_n|^{1/n} = 1\cdot L = L$ since $n^{1/n}\to 1$ ; so $R$ is unchanged. Equality of $f'$ with the term-by-term derivative follows from uniform convergence of the differentiated series on each compact subdisk $|x-c|\le \rho < R$ (Weierstrass M-test), justifying interchange of limit and derivative.

∎

JH

Jacques Hadamard

1865–1963

Though primarily a pure mathematician, Hadamard's 1888 thesis crystallized the formula now bearing his name (with Cauchy), giving the exact radius of convergence via the $\limsup$ of $|a_n|^{1/n}$ — the first fully general statement, valid even when the ratio $|a_n/a_{n+1}|$ has no limit. Hadamard also proved the Prime Number Theorem (1896, independently of de la Vallée Poussin), worked extensively on partial differential equations and the calculus of variations, and articulated the influential notion of a well-posed problem. His insight that the growth rate of coefficients alone dictates convergence transformed power series from a computational tool into a structural theory linking real analysis to complex function theory.

🧠 Intuition

If $|x-c| < R$ , the powers shrink fast enough to crush whatever growth $|a_n|$ has — the terms decay geometrically, like a convergent geometric series with ratio $|x-c|/R < 1$ .
If $|x-c| > R$ , the powers grow faster than the coefficients can be tamed, the terms blow up, and the series has no chance.

The number $|a_n|^{1/n}$ measures the "average geometric growth rate per term" of the coefficients; its reciprocal is exactly the distance at which the powers can offset that growth. This is why $R$ depends only on the coefficients, never on $x$ .

Figure

Convergence behavior on the real line: absolute convergence inside, divergence outside, indeterminate at the endpoints.

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ExampleThe Exponential Series — Infinite Radius

Given: $\displaystyle f(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$ , so $a_n = 1/n!$ , $c = 0$ .

Step 1 — Ratio form $[4]$ .

\left|\frac{a_n}{a_{n+1}}\right| = \frac{1/n!}{1/(n+1)!} = \frac{(n+1)!}{n!} = n+1.

Step 2 — Take the limit.

R = \lim_{n\to\infty}(n+1) = \infty.

Step 3 — Interpret. The series converges absolutely for every real (and complex) $x$ . There are no endpoints to test since $R=\infty$ .

\boxed{R = \infty, \qquad \text{interval of convergence } = (-\infty,\infty)}

This is consistent with $f(x) = e^x$ being defined and analytic everywhere.

ExampleEndpoints Differ — A Logarithmic Series

Given: $\displaystyle f(x) = \sum_{n=1}^\infty \frac{x^n}{n}$ , so $a_n = 1/n$ , $c = 0$ .

Step 1 — Radius via ratio.

R = \lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right| = \lim_{n\to\infty}\frac{1/n}{1/(n+1)} = \lim_{n\to\infty}\frac{n+1}{n} = 1.

So the open interval is $(-1,1)$ .

Step 2 — Test the right endpoint $x = 1$ . The series becomes $\sum 1/n$ , the harmonic series, which diverges.

Step 3 — Test the left endpoint $x = -1$ . The series becomes $\sum (-1)^n/n$ , the alternating harmonic series, which converges (conditionally, by the alternating series test).

\boxed{R = 1, \qquad \text{interval of convergence } = [-1, 1)}

This illustrates that the two endpoints can — and often do — behave completely differently. (Indeed $f(x) = -\ln(1-x)$ , whose singularity at $x=1$ explains the divergence there.)

⚠️ Common mistake

The ratio test is not always applicable. The formula $R = \lim |a_n/a_{n+1}|$ requires the limit to exist. For series with gaps (e.g. only even powers) or oscillating coefficient ratios, the limit may fail. The Cauchy–Hadamard formula $[3]$ with the limsup is the universally correct tool. Also note: convergence at $|x-c|=R$ tells you nothing about the other endpoint or about absolute vs. conditional convergence — each boundary point must be checked by hand.

CounterexampleRatio Test Failure on a Lacunary Series

Consider

f(x) = \sum_{n=0}^\infty x^{2^n} = x + x^2 + x^4 + x^8 + x^{16} + \cdots

Here the coefficient of $x^k$ is $a_k = 1$ if $k$ is a power of $2$ and $a_k = 0$ otherwise. The ratio $|a_k/a_{k+1}|$ is undefined infinitely often (division by zero), so $[4]$ cannot be used. But Cauchy–Hadamard $[3]$ still works: $\limsup_k |a_k|^{1/k} = 1$ (since $|a_k|^{1/k}$ is either $1$ or $0$ , and equals $1$ infinitely often), giving $R = 1$ . Moreover, the unit circle $|x|=1$ is a natural boundary: $f$ cannot be analytically continued past it at any boundary point — a phenomenon invisible to the radius alone.

connection

Power series sit at the crossroads of several theories:

A function equal to its Taylor Series on a neighborhood is precisely an analytic function; power series are the local building blocks of Analytic Functions.
Over $\mathbb{C}$ , the radius of convergence equals the distance from the center to the nearest singularity, tying $R$ to Complex Analysis and the residue theory of poles and branch points.
Generating Functions in combinatorics are power series whose coefficients encode counting sequences; $R$ controls the asymptotic growth of those sequences via $a_n \sim R^{-n}$ .
The Frobenius method solves linear Ordinary Differential Equations by assuming power-series solutions and recovering recurrences for the $a_n$ .
Uniform Convergence on compact subdisks (via the Weierstrass M-test) is the engine that legitimizes term-by-term differentiation $[6]$ and integration.

🔬 Where this appears

⏳ History

- 1668–1671 — Nicolaus Mercator and Isaac Newton develop series for

\ln(1+x)

and the generalized binomial series, pioneering power series as a computational engine.

- 1715 — Brook Taylor publishes the Taylor expansion (anticipated by Gregory and Newton), expressing functions as power series in a single point.

- 1748 — Euler's Introductio in analysin infinitorum treats power series formally and freely, deriving countless identities (often without convergence concerns).

- 1821 — Augustin-Louis Cauchy's Cours d'analyse rigorously defines convergence and introduces the root test, laying groundwork for what becomes the Cauchy–Hadamard formula.

- 1826 — Niels Henrik Abel proves Abel's theorem on boundary continuity (

\sum a_n x^n \to \sum a_n

as

x\to R^-

when the endpoint series converges) and corrects loose 18th-century practice.

- 1851–1860s — Weierstrass builds analytic function theory on power series as the primitive objects, proving uniform convergence properties and the M-test that justify term-by-term operations.

- 1888 — Jacques Hadamard, in his doctoral thesis, states the general formula

1/R = \limsup |a_n|^{1/n}

, valid without assuming the ratio limit exists.

- 1892 — Hadamard's gap theorem on natural boundaries shows lacunary series cannot be continued past

|x|=R

.

- 20th c. — The theory generalizes to power series over

\mathbb{C}

, Banach spaces (formal/analytic functional calculus), and

p

-adic fields, with the radius concept persisting throughout.

open-problem

Several frontiers remain genuinely deep even beyond undergraduate scope:

Boundary behavior is largely undetermined in general: while Abel's theorem handles convergent endpoints and Tauberian theorems give partial converses, classifying exactly which boundary points of a given series converge can be intractable. Carleson's theorem (1966) on almost-everywhere convergence of Fourier series is a celebrated hard result in this circle of ideas.
Divergent yet useful series. Many physical expansions have $R = 0$ (asymptotic series). Borel summation and resurgence theory (Écalle) extract meaning from such series; understanding which divergent series are "resummable" and how is active research.
Natural boundaries. Determining whether a power series with a given coefficient pattern has the unit circle as a natural boundary — and the fine structure of singularities on it — connects to the Fabry gap theorem and remains delicate.
Several complex variables. The single radius $R$ becomes a domain of convergence with intricate (logarithmically convex) geometry; the theory of domains of holomorphy and pseudoconvexity is a rich modern subject.

Power Series and Radius of Convergence

Formal Statement

Why It Holds

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also