Power Series

A power series is an infinite series of the form $\sum_{n = 0}^{\infty} a_{n} (x - c)^{n}$ , a "polynomial of infinite degree" centered at $c$ . Key result: every power series has a radius of convergence $R$ (possibly $0$ or $\infty$ ) inside which it converges absolutely and defines an infinitely differentiable function. Where it applies: representing functions, solving ODEs, complex analysis, generating functions, numerical computation. Who/when: Newton (~1665), Taylor (1715), Cauchy and Abel (1820s) gave the rigorous convergence theory.

A power series centered at $c$ is the expression $\sum_{n = 0}^{\infty} a_{n} (x - c)^{n}$ , where the $a_{n}$ are fixed coefficients and $x$ is a variable. For each value of $x$ this becomes an ordinary infinite series that may converge or diverge; the set of $x$ for which it converges is always an interval (or disk, in the complex case) symmetric about $c$ . Power series matter because the functions we care about most — $e^{x}$ , $sin x$ , $ln (1 + x)$ , and solutions of differential equations — can be represented by them, turning transcendental analysis into algebra of coefficients. A genuinely surprising fact is rigidity: a function representable by a power series is determined globally by its derivatives at the single point $c$ , so two such functions agreeing on any tiny interval must agree everywhere they are both defined.

The Core Insight

A power series is the bridge between local data (the coefficients $a_{n}$ , which encode derivatives at one point) and a function on an interval. Convergence is governed by a single number $R$ , the radius of convergence, determined entirely by how fast $∣ a_{n} ∣$ grows. Inside $R$ the series behaves like an infinitely-long polynomial: you may differentiate, integrate, add, and multiply term-by-term. At the boundary $∣ x - c ∣ = R$ anything can happen.

Formal Statement

Power Series

A power series centered at $c$ with coefficients ${a_{n}}_{n \geq 0}$ is the series

n = 0 \sum \infty a_{n} (x - c)^{n} = a_{0} + a_{1} (x - c) + a_{2} (x - c)^{2} + \dots

⟳this is the definition; each term is a coefficient

a_{n} \in R

(or

C

) times a power of the displacement

x - c

from the center

c

Here $x$ is the variable, $c$ is the center, $a_{n}$ is the $n$ -th coefficient, and $n$ ranges over the non-negative integers. The series defines a function $f (x)$ on the set of $x$ where it converges, with $f (c) = a_{0}$ always (every other term vanishes at $x = c$ ).

Radius and Interval of Convergence

The radius of convergence $R \in [0, \infty]$ is the number such that the series [1] converges absolutely for $∣ x - c ∣ < R$ and diverges for $∣ x - c ∣ > R$ . It is given by the Cauchy–Hadamard formula

\frac{1}{R} = n \to \infty lim sup ∣ a_{n} ∣^{1/ n}

⟳apply the root test to

\sum ∣ a_{n} (x - c)^{n} ∣

; the series converges when

lim sup ∣ a_{n} (x - c)^{n} ∣^{1/ n} = ∣ x - c ∣ lim sup ∣ a_{n} ∣^{1/ n} < 1

with conventions $1/0 = \infty$ and $1/\infty = 0$ . The interval of convergence is $(c - R, c + R)$ together with whichever endpoints converge — these must be tested separately.

Cauchy–Hadamard Theorem

For the power series $\sum a_{n} (x - c)^{n}$ with $R$ defined by [2]:

The series converges absolutely for every $x$ with $∣ x - c ∣ < R$ .
The series diverges for every $x$ with $∣ x - c ∣ > R$ .
On any closed sub-interval $[c - r, c + r]$ with $r < R$ , convergence is uniform.

No general claim is made for $∣ x - c ∣ = R$ .

Ratio Test for Radius

If the limit exists, the radius is also given by

R = n \to \infty lim \frac{a _{n}}{a _{n + 1}}

⟳apply the ratio test to the terms

u_{n} = a_{n} (x - c)^{n}

; convergence requires

lim ∣ u_{n + 1} / u_{n} ∣ = ∣ x - c ∣ lim ∣ a_{n + 1} / a_{n} ∣ < 1

This is easier to apply when ratios simplify, but [2] always works whereas [3] can fail when the limit does not exist.

Why It Holds / Derivation

Existence of a Radius via Abel's Lemma

Step 1 — A convergence-at-one-point lemma. Suppose

\sum a_{n} (x_{0} - c)^{n}

converges for some particular

x_{0} \neq = c

. Then its terms are bounded: there is

M

with

∣ a_{n} (x_{0} - c)^{n} ∣ \leq M

for all

n

, since terms of a convergent series tend to

0

. Step 2 — Comparison with a geometric series. For any

x

with

∣ x - c ∣ < ∣ x_{0} - c ∣

, set

t = \frac{∣ x - c ∣}{∣ x _{0} - c ∣} < 1

. Then

∣ a_{n} (x - c)^{n} ∣ = ∣ a_{n} (x_{0} - c)^{n} ∣ \cdot t^{n} \leq M t^{n}

⟳factor

(x - c)^{n} = (x_{0} - c)^{n} (\frac{x - c}{x _{0} - c})^{n}

and bound the first factor by

M

using Step 1.

Step 3 — Conclusion. Since

\sum_{n} M t^{n} = \frac{M}{1 - t} < \infty

, the comparison test gives absolute convergence at

x

. Thus convergence at

x_{0}

forces convergence at every closer point. Step 4 — Define $R$ . Let

R = sup {∣ x_{0} - c ∣ : \sum a_{n} (x_{0} - c)^{n} converges}

. By Step 3, all points strictly inside radius

R

converge; points strictly outside cannot converge (else they would force convergence at a point beyond

R

, contradicting the sup). This

R

coincides with [2].

■

Term-by-Term Differentiation and Integration

If $f (x) = \sum_{n = 0}^{\infty} a_{n} (x - c)^{n}$ has radius $R > 0$ , then $f$ is infinitely differentiable on $(c - R, c + R)$ , and the derivative and antiderivative series

f^{'} (x) = n = 1 \sum \infty n a_{n} (x - c)^{n - 1}, \int f d x = C + n = 0 \sum \infty \frac{a _{n}}{n + 1} (x - c)^{n + 1}

⟳uniform convergence on compact sub-intervals (Cauchy–Hadamard part 3) permits interchange of limit and derivative/integral;

lim sup (n ∣ a_{n} ∣)^{1/ n} = lim sup ∣ a_{n} ∣^{1/ n}

shows the radius is unchanged.

both have the same radius of convergence $R$ .

Taylor Coefficients are Forced

If $f (x) = \sum_{n = 0}^{\infty} a_{n} (x - c)^{n}$ on $(c - R, c + R)$ , then differentiating [1] repeatedly and evaluating at $x = c$ gives

a_{n} = \frac{f ^{(n)} ( c )}{n !}

⟳apply [5]

n

times; all terms with power

< n

vanish under differentiation, all with power

> n

vanish at

x = c

, leaving only

n! a_{n}

So a function has at most one power series representation at a given center — its Taylor series.

Niels Henrik Abel | 1802–1829

The Norwegian mathematician Abel, in his 1826 paper on the binomial series, gave the first fully rigorous treatment of power-series convergence at a time when "summing divergent series" was rampant. He famously complained that "divergent series are the invention of the devil." His Abel's theorem — that if $\sum a_{n} R^{n}$ converges then $f (x) \to \sum a_{n} R^{n}$ as $x \to R^{-}$ — rescued endpoint behavior and justified results like $ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \dots$ . Abel died of tuberculosis at 26, days before a letter arrived offering him a professorship in Berlin.

Building Intuition

Think of a power series as a dial of weights $a_{n}$ tuning how strongly each power $(x - c)^{n}$ contributes. Near the center $x = c$ , the low powers dominate and the series looks like its first few terms — a polynomial approximation. As you move away from $c$ , higher powers grow and the question becomes whether the coefficients shrink fast enough to tame them. The radius $R$ is exactly the tipping point: a competition between coefficient decay ( $∣ a_{n} ∣^{1/ n}$ ) and distance growth ( $∣ x - c ∣$ ). The geometric series $\sum x^{n} = \frac{1}{1 - x}$ is the prototype — it converges for $∣ x ∣ < 1$ and blows up exactly where the function it represents has a singularity, at $x = 1$ . This is the deep moral: the radius of convergence is the distance to the nearest singularity (visible clearly in the complex plane).

Convergence behavior of a power series on the real line

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Worked Examples

Finding the Interval of Convergence

Given: the series

n = 1 \sum \infty \frac{( x - 2 ) ^{n}}{n 3 ^{n}}

. Find

R

and the full interval of convergence. Step 1 — Apply the ratio test for the radius. Here

a_{n} = \frac{1}{n 3 ^{n}}

. Compute

\frac{a _{n}}{a _{n + 1}} = \frac{( n + 1 ) 3 ^{n + 1}}{n 3 ^{n}} = 3 \cdot \frac{n + 1}{n} \to 3.

R = 3

and the open interval is

(2 - 3, 2 + 3) = (- 1, 5)

. Step 2 — Test the left endpoint $x = - 1$ . Then

(x - 2)^{n} = (- 3)^{n}

, giving

n = 1 \sum \infty \frac{( - 3 ) ^{n}}{n 3 ^{n}} = n = 1 \sum \infty \frac{( - 1 ) ^{n}}{n},

the alternating harmonic series, which converges. Step 3 — Test the right endpoint $x = 5$ . Then

(x - 2)^{n} = 3^{n}

, giving

n = 1 \sum \infty \frac{3 ^{n}}{n 3 ^{n}} = n = 1 \sum \infty \frac{1}{n},

the harmonic series, which diverges. Answer:

R = 3, interval of convergence = [- 1, 5)

Summing and Differentiating a Series

Given: Starting from the geometric series

\frac{1}{1 - x} = n = 0 \sum \infty x^{n}

for

∣ x ∣ < 1

, evaluate

S = n = 1 \sum \infty \frac{n}{2 ^{n}}

. Step 1 — Differentiate term-by-term (valid for $∣ x ∣ < 1$ by [5]).

\frac{d}{d x} \frac{1}{1 - x} = \frac{1}{( 1 - x ) ^{2}} = n = 1 \sum \infty n x^{n - 1} .

Step 2 — Multiply by $x$ to match the desired powers.

\frac{x}{( 1 - x ) ^{2}} = n = 1 \sum \infty n x^{n} .

Step 3 — Substitute $x = \frac{1}{2}$ (inside $∣ x ∣ < 1$ , so legal).

S = n = 1 \sum \infty \frac{n}{2 ^{n}} = \frac{1/2}{( 1 - 1/2 ) ^{2}} = \frac{1/2}{1/4} = 2.

Answer:

n = 1 \sum \infty \frac{n}{2 ^{n}} = 2

Common Misconceptions

Convergence does not equal representation. A function can be infinitely differentiable, have a Taylor series that converges everywhere, and yet that series may not equal the original function. Convergence of the Taylor series is necessary but not sufficient for

f

to be analytic. You must check that the remainder term

R_{n} (x) \to 0

, not merely that the series converges. Endpoints are NOT determined by $R$ . The interval can be open, half-open, or closed. Always test

x = c \pm R

separately — the ratio/root tests are silent there because they give the boundary case

L = 1

A Smooth Function Whose Taylor Series Lies

Define

f (x) = {e^{- 1/ x^{2}}, 0, x \neq = 0 x = 0.

Every derivative

f^{(n)} (0) = 0

(the exponential decay beats every polynomial). Therefore its Maclaurin series is

n = 0 \sum \infty \frac{f ^{(n)} ( 0 )}{n !} x^{n} = 0 + 0 \cdot x + 0 \cdot x^{2} + \dots = 0,

which converges (to

0

) for all

x

— yet

f (x) \neq = 0

for any

x \neq = 0

. The series converges everywhere but represents

f

only at the single point

x = 0

. Such functions are smooth (

C^{\infty}

) but not analytic.

Connections

Power series are the analytic backbone connecting several areas. They specialize to Taylor Series when coefficients come from derivatives via [6], and to Maclaurin Series when $c = 0$ . The Geometric Series is the foundational example and the engine for deriving many others by substitution and term-by-term operations. In Complex Analysis, a function is holomorphic precisely when it is locally a power series, and the radius equals the distance to the nearest singularity in $C$ . They generalize to Laurent Series (allowing negative powers) and connect to Fourier Series as another way to expand functions in a basis. The convergence theory rests on the Ratio Test, Root Test, and Uniform Convergence, and the boundary behavior is governed by Abel's Theorem.

Real-World Applications

Numerical computation of transcendental functions. Calculators and libraries (e.g., the C math.h, CORDIC variants) evaluate $sin$ , $cos$ , $exp$ , $ln$ via truncated power series with controlled remainder bounds.
Solving differential equations. The Frobenius method finds power-series solutions to ODEs like Bessel's and Legendre's equations, producing the special functions of mathematical physics.
Physics — perturbation theory. Quantum mechanics expands energies and wavefunctions as power series in a coupling constant $λ$ ; the small- $λ$ regime gives the Stark and Zeeman effect corrections.
Combinatorics — generating functions. Sequences (Fibonacci, Catalan, partitions) are encoded as coefficients of power series; algebraic manipulation of the series yields closed forms and recurrences.
Signal processing — the z-transform. Discrete-time systems are analyzed through $\sum x [n] z^{- n}$ , a power series whose region of convergence determines stability and causality.
Economics and finance. Present-value calculations of perpetuities and growing annuities are geometric power series; bond pricing sums discounted cash flows.
Probability — moment and probability generating functions. $E [z^{X}] = \sum p_{k} z^{k}$ encodes a distribution; derivatives at $z = 1$ recover moments.

Historical Development

~1665 — Isaac Newton develops the generalized binomial series $(1 + x)^{α} = \sum (n α) x^{n}$ and uses infinite series freely to compute areas and invert functions.
1668 — Nicholas Mercator publishes the series for $ln (1 + x)$ .
1671 — James Gregory and (independently) Newton obtain the arctangent series, later used to compute $π$ .
1715 — Brook Taylor publishes Methodus Incrementorum, formalizing what becomes Taylor's theorem, though without convergence considerations.
1742 — Colin Maclaurin popularizes the special case centered at $0$ in his Treatise of Fluxions.
1748 — Leonhard Euler's Introductio treats series as a central object, deriving $e^{i x} = cos x + i sin x$ and many identities — formally, with little rigor.
1821 — Augustin-Louis Cauchy's Cours d'analyse introduces rigorous limits and the root test, giving (with Hadamard later) the radius formula [2].
1826 — Niels Abel proves the binomial series rigorously and establishes Abel's theorem on endpoint convergence.
1892 — Jacques Hadamard's thesis cements the $lim sup$ form of [2], now called the Cauchy–Hadamard theorem.

Subtleties & Limits

The hardest questions concern the boundary $∣ x - c ∣ = R$ and the gap between convergence and analytic representation.

Boundary behavior is delicate. A series may converge at some boundary points and diverge at others; characterizing the convergence set on the circle $∣ z - c ∣ = R$ (in the complex case) connects to deep results like the Fatou and Carleson theorems on radial and tangential limits.
Divergent-yet-useful series. Asymptotic series (e.g., for $\int_{0}^{\infty} \frac{e ^{- t}}{1 + x t} d t$ ) diverge for every $x \neq = 0$ yet give superb finite-term approximations — formalized by Borel summation and the theory of resurgence (Écalle), still active research.
Lacunary series and natural boundaries. Series like $\sum z^{2^{n}}$ converge on $∣ z ∣ < 1$ but cannot be analytically continued past the boundary circle at all (Hadamard's gap theorem) — the unit circle is a "wall." Determining when such natural boundaries occur remains subtle.
Multivariable and operator-valued series raise convergence questions (domains of convergence are no longer disks; "Reinhardt domains") that are genuinely harder and not fully classified.

Power Series

The Core Insight

Formal Statement

Power Series

Radius and Interval of Convergence

Cauchy–Hadamard Theorem

Ratio Test for Radius

Why It Holds / Derivation

Existence of a Radius via Abel's Lemma

Term-by-Term Differentiation and Integration

Taylor Coefficients are Forced

Niels Henrik Abel | 1802–1829

Building Intuition

Convergence behavior of a power series on the real line

Worked Examples

Finding the Interval of Convergence

Summing and Differentiating a Series

Common Misconceptions

A Smooth Function Whose Taylor Series Lies

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also