Power Series

Motivation: why this matters in quant finance

A power series is an infinite polynomial

\sum a_n x^n

. The Taylor series is the special case where

a_n = f^{(n)}(0)/n!

, but power series appear in quant finance in contexts far beyond Taylor approximation:

Moment generating functions (MGFs) and characteristic functions are power series (or their complex analogues) that encode the entire distribution of a random variable. The MGF of a normal $\mathcal{N}(\mu, \sigma^2)$ is $M(t) = e^{\mu t + \sigma^2 t^2/2}$ , which when expanded as a power series in $t$ yields all moments: $\mathbb{E}[X^n] = M^{(n)}(0)$ .
Generating functions for discrete distributions (probability generating functions, $z$ -transforms) are power series in a formal variable. They are used in credit portfolio models (generating functions for loss distributions), in counting combinatorial structures (lattice models), and in recursive option pricing.
The radius of convergence of a power series tells you the domain where your approximation is valid — when a pricing approximation breaks down, it is often because you have left the radius of convergence.

Definition

A power series centred at

a

is:

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

where $\{c_n\}$ is a sequence of coefficients and $x$ is the variable. When $a = 0$ :

\sum_{n=0}^{\infty} c_n x^n

A Taylor series is a power series with

c_n = f^{(n)}(a)/n!

. But a power series can be defined abstractly by its coefficients

\{c_n\}

without reference to any underlying function.

Radius of convergence

Definition

Every power series

\sum c_n x^n

has a radius of convergence

R \in [0, \infty]

such that:

The series converges absolutely for $|x| < R$ .
The series diverges for $|x| > R$ .
At $|x| = R$ , convergence must be checked case by case.

Computing $R$

Ratio test:

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

(when the limit exists).

Root test (Hadamard formula):

1/R = \limsup_{n \to \infty} |c_n|^{1/n}

See Convergence of Sequences and Series for details on these tests.

Key examples

Power series	Coefficients	Radius $R$
$e^x = \sum x^n/n!$	$c_n = 1/n!$	$\infty$
$\ln(1+x) = \sum (-1)^{n+1}x^n/n$	$c_n = (-1)^{n+1}/n$	$1$
$1/(1-x) = \sum x^n$	$c_n = 1$	$1$
$(1+x)^\alpha = \sum \binom{\alpha}{n}x^n$	$c_n = \binom{\alpha}{n}$	$1$

Finance implication: The approximation

\ln(1+R) \approx R - R^2/2

(from the Taylor/power series of

\ln(1+x)

) is valid for

|R| < 1

— returns less than 100%. For daily equity returns (

|R| \approx 0.01

), this is extremely safe. For cumulative returns over long horizons or in crises,

|R|

can approach or exceed 1, and the approximation fails.

Properties

Term-by-term differentiation and integration

Within the radius of convergence, a power series can be differentiated and integrated term by term:

\frac{d}{dx}\sum c_n x^n = \sum n c_n x^{n-1}, \qquad \int \sum c_n x^n\,dx = \sum \frac{c_n}{n+1}x^{n+1} + C

Both resulting series have the same radius of convergence

R

Finance application: The MGF

M(t) = \sum_{n=0}^{\infty} \frac{\mathbb{E}[X^n]}{n!}\,t^n

can be differentiated term by term to recover moments:

M'(0) = \mathbb{E}[X]

M''(0) = \mathbb{E}[X^2]

, etc. This is valid within the radius of convergence of the MGF — which may be finite (e.g., for heavy-tailed distributions, the MGF may not exist for

t > 0

Uniqueness of coefficients

If $\sum a_n x^n = \sum b_n x^n$ for all $x$ in some interval around 0, then $a_n = b_n$ for all $n$ . Power series representations are unique.

This means: if you find the power series of a function by any method (Taylor's formula, algebraic manipulation, differential equation), the coefficients are guaranteed to be the same.

Multiplication and composition

If $f(x) = \sum a_n x^n$ and $g(x) = \sum b_n x^n$ both converge for $|x| < R$ , then:

Product (Cauchy product):

f(x)g(x) = \sum_{n=0}^{\infty}\left(\sum_{k=0}^{n} a_k b_{n-k}\right) x^n

Composition

f(g(x))

is valid when

|g(x)| < R_f

(the inner function must stay within the radius of the outer series).

Moment generating functions as power series

Definition and connection

The moment generating function of a random variable

X

is:

M_X(t) = \mathbb{E}[e^{tX}] = \sum_{n=0}^{\infty} \frac{\mathbb{E}[X^n]}{n!}\,t^n

This is a power series in $t$ with coefficients $c_n = \mathbb{E}[X^n]/n!$ . It converges in a neighbourhood of $t = 0$ if all moments exist and grow slowly enough.

For the normal distribution

X \sim \mathcal{N}(\mu, \sigma^2)

M_X(t) = e^{\mu t + \sigma^2 t^2/2}

This converges for all $t \in \mathbb{R}$ (radius $R = \infty$ ), reflecting the fact that the normal distribution has finite moments of all orders and light (Gaussian) tails.

When the MGF does not exist

For heavy-tailed distributions, the MGF may have

R = 0

— it only converges at

t = 0

. For example, the Student's

t

distribution with

\nu

degrees of freedom has MGF that diverges for all

t \neq 0

when

\nu

is small. The characteristic function

\varphi(t) = \mathbb{E}[e^{itX}]

(with

i = \sqrt{-1}

) always exists and serves as a replacement — see Fourier Series and Transforms.

Cumulant generating function

The cumulant generating function

K(t) = \ln M(t)

is also a power series:

K(t) = \sum_{n=1}^{\infty} \frac{\kappa_n}{n!}\,t^n

where

\kappa_n

are the cumulants:

\kappa_1 = \mu

\kappa_2 = \sigma^2

\kappa_3 =

skewness

\times \sigma^3

\kappa_4 =

excess kurtosis

\times \sigma^4

For the normal distribution,

K(t) = \mu t + \sigma^2 t^2/2

, so

\kappa_n = 0

for

n \geq 3

. Non-zero higher cumulants measure departure from normality. In the Edgeworth expansion (a refinement of the CLT), these cumulants appear as correction terms to the Gaussian approximation.

Probability generating functions

For a discrete random variable $N$ taking non-negative integer values:

G_N(z) = \mathbb{E}[z^N] = \sum_{k=0}^{\infty} \mathbb{P}(N = k)\,z^k

This is a power series in $z$ with coefficients $p_k = \mathbb{P}(N = k)$ .

Finance application — credit portfolio losses: In the CreditRisk+ model, the number of defaults

N

in a portfolio follows a Poisson mixture. The probability generating function of the total loss is:

G_L(z) = \prod_{i} G_{L_i}(z)

where the product is over independent obligors. The coefficients of

G_L(z)

give the loss distribution. This is computed via power series multiplication — exactly the Cauchy product rule.

Analytic continuation

A function defined by a power series on

|x| < R

may extend to a larger domain via analytic continuation — finding another power series (centred at a different point) that agrees on an overlap region.

Finance application: The Black-Scholes formula, originally derived for

S > 0

\sigma > 0

T > 0

, can be extended to edge cases (e.g.,

T \to 0

\sigma \to 0

) by analytic continuation of the underlying power series expressions. Understanding the radius of convergence tells you which limiting cases are well-behaved and which require special treatment.

Common confusions and pitfalls

"A power series converges everywhere or nowhere." No. Most power series have a finite radius of convergence

R

, converging inside the disk

|x| < R

and diverging outside. Only special series like

e^x

converge everywhere (

R = \infty

"If the MGF doesn't exist, the distribution has no moments." Not necessarily. The MGF can fail to converge (for any

t \neq 0

) even when all moments exist, if the moments grow too fast. Conversely, some distributions with finite low-order moments but infinite higher-order moments have MGFs that converge on a bounded interval. The characteristic function is the universal alternative.

"The radius of convergence tells you where the function is undefined." Not exactly.

1/(1-x) = \sum x^n

has

R = 1

, and indeed

1/(1-x)

has a pole at

x = 1

. But the function is perfectly well-defined for

x > 1

— the power series just cannot represent it there. The radius detects the nearest singularity in the complex plane, which may not be visible on the real line.

Where this goes next

Power series connect to:

Taylor Series: Taylor series are power series with $c_n = f^{(n)}(a)/n!$ . The radius of convergence of the Taylor series determines where the Taylor approximation is valid.
Fourier Series and Transforms: The characteristic function $\varphi(t) = \mathbb{E}[e^{itX}]$ is a "power series with complex argument" and exists even when the MGF does not.
Convergence of Sequences and Series: The ratio and root tests determine the radius of convergence.
Normal Distribution and Log-Normal Distribution: The MGF $e^{\mu t + \sigma^2 t^2/2}$ and the moment formula $\mathbb{E}[e^{nY}] = e^{n\mu + n^2\sigma^2/2}$ are derived from power series.
Asymptotic Expansions: When power series converge too slowly or not at all, asymptotic expansions provide useful approximations.

References

Stewart, J. (2008). Single Variable Calculus: Early Transcendentals (6th ed.). Thomson Brooks/Cole. Ch. 11 Sections 11.8-11.9 (Power Series and Representations of Functions as Power Series) for radius of convergence and algebraic manipulation.

Power Series

Motivation: why this matters in quant finance

Definition

Radius of convergence

Definition

Computing RRR

Key examples

Properties

Term-by-term differentiation and integration

Uniqueness of coefficients

Multiplication and composition

Moment generating functions as power series

Definition and connection

When the MGF does not exist

Cumulant generating function

Probability generating functions

Analytic continuation

Common confusions and pitfalls

Where this goes next

References

Computing $R$