Power Series as Functions: Limits, Exponentials, and Taylor's Theorem

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Introduction

Definition. Complex sequence A sequence of complex numbers is a function $a: \mathbb{N} \to \mathbb{C}$. We usually denote it by $(a_n)_{n \in \mathbb{N}}$ or simply $(a_n)$, where $a_n := a(n)$. The value $a_1$ is called the first term of the sequence, $a_2$ the second term, and in general $a_n$ the n-th term of the sequence.

Definition. Convergent complex sequence. A complex sequence $(a_n)_{n \in \mathbb{N}}$ is said to converge to a complex number $L \in \mathbb{C}$ if for every $\varepsilon > 0$ there exists an integer $N \in \mathbb{N}$ such that for all $n \geq N$ one has $|a_n - L| < \varepsilon$. In this case we write $\lim_{n \to \infty} a_n = L$ or $a_n \to L$ as $n \to \infty$, and L is called the limit of the sequence $(a_n)_{n \in \mathbb{N}}$.

Definition. Cauchy sequence. A complex sequence $(a_n)_{n \in \mathbb{N}}$ is called a Cauchy sequence if for every $\varepsilon > 0$ there exists an integer $N \in \mathbb{N}$ such that for all $n, m \geq N$ one has $|a_n - a_m| < \varepsilon$.

Definition. Series and partial sums.Let $(a_n)_{n \in \mathbb{N}}$ be a complex sequence. For each n $\in \mathbb{N}$, the finite sum $s_n := a_1 + a_2 + \cdots + a_n = \sum_{k=1}^n a_k$ is called the n-th partial sum of the (infinite) series $\sum_{k=1}^\infin a_k$ which we also denote simply by $\sum a_n$ when the index is clear from the context.

Definition. Convergent series. The series $\sum_{n=1}^{\infty} a_n$ is said to converge to the sum $s \in \mathbb{C}$ if the sequence of partial sums $(s_n)_{n \in \mathbb{N}}$ defined by $s_n = a_1 + a_2 + \cdots + a_n = \sum_{k=1}^n a_k$ converges to s, that is, $\lim_{n \to \infty} s_n = s$. In this case we write $s := \sum_{n=1}^\infin a_n$. If the sequence $(s_n)_{n \in \mathbb{N}}$ does not converge, we say that the series $\sum_{n=1}^{\infty} a_n$ diverges (or does not converge).

Definition. A complex power series centered at 0 in the variable z is a series of the form $a_0 + a_1z + a_2z^2 + \cdots = \sum_{n=0}^\infty a_n z^n$ with coefficients $a_i \in \mathbb{C}$

Definition. A complex power series centered at a complex number $a \in \mathbb{C} $ is an infinite series of the form: $\sum_{n=0}^\infty a_n (z - a)^n,$ where each $a_n \in \mathbb{C}$ is a coefficient, z is a complex variable, and $(z - a)^n$ is the nth power about the center.

Theorem. Given a power series $\sum_{n=0}^\infty a_n z^n$, there exists a unique value R, $0 \le R \le \infin$ (called the radius of convergence) such that:

1. For any z with |z| < R (inside the circle), the series $\sum_{n=0}^\infty a_n z^n$ converges absolutely (this is a “green light” zone).
2. For any z with |z| > R, the series diverges (this is a “red light” zone).

   On the Circle (|z| = R), this theorem gives no information. This is the yellow light zone —the series could converge or diverge.

Differentiability of Power Series. If $f(z) = \sum_{n=0}^{\infty} a_nz^n$ for |z| < R (R > 0), then f is analytic on B(0; R) and $f'(z) = \sum_{n=1}^{\infty} na_nz^{n-1}$ for |z| < R.

Power Series as Functions: Limits, Exponentials, and Taylor’s Theorem

At its core, a power series is not a static algebraic expression; it is a dynamic process —a sequence of polynomials that approximate a function.

Power series as a limit. A power series centered at $z=0$ $f(z) = \sum_{n=0}^{\infty} a_nz^n$ defines a function by the limit of its partial sums: $f(z)= \lim_{N \to \infty} S_N(z), \text{ where } S_N(z)=\sum_{n=0}^{N}a_nz^n$.

1. The domain of the function $f(z)$ is precisely the set of complex numbers $z$ for which this limit exists (converges).
2. Radius of convergence. By the Root or Ratio Test, there exists a unique $R \in [0, \infty]$ (radius of convergence) such that:
   (i) the series converges absolutely for all |z| < R.
   (ii) the series diverges for all |z| > R. (iii) on the boundary circle |z| = R, the convergence must be checked case-by-case.
3. At the origin z = 0, for any finite integer $N \ge 0$, the partial sum evaluated at $z=0$ is: $S_N(0)=a_0 \cdot 0^0 + a_1 \cdot 0^1+\dots +a_N \cdot0^N$.

In power-series contexts, we universally adopt the convention that $z^0 = 1$ for all z, including z = 0. This convention eliminates the usual $0^0$ indeterminacy, makes the constant term well defined, and the algebraic notation $\sum a_n z^n$ perfectly rigorous without needing piecewise definitions. Hence, $S_N(0) = a_0, \forall N$, so the limit is trivial $N \to \infty$ is also $a_0, f(0)=\lim_{N \to \infty} S_N(0) = a_0$. A verbose restatement of the fact is as follows $f(z) = \begin{cases} a_0, &z = 0 \\\\ \sum_{n=0}^{\infty} a_nz^n, &z \ne 0 \end{cases}$

Case Study: The Complex Exponential

To see these principles in action, let us define a brand-new function $f(z)$ purely using a power series: $f(z) = \sum_{n=0}^{\infty} \frac{z^n}{n!}$. Our first step is to find out where this function is well-defined, i.e., for which complex numbers z the series converges.

We use the Ratio Test for absolute convergence, which states that a series converges absolutely if the limit of the magnitude of the ratio of consecutive terms is less than $1$.

Let $a_n(z) = \frac{z^n}{n!}$. For any fixed $z \in \mathbb{C}$: $\left| \frac{a_{n+1}(z)}{a_n(z)} \right| = \left| \frac{\frac{z^{n+1}}{(n+1)!}}{\frac{z^n}{n!}} \right| = \left| \frac{z^{n+1}}{(n+1)!} \cdot \frac{n!}{z^n} \right| = \left| \frac{z}{n+1} \right|$

Now, we take the limit as $n \to \infty$.

$L = \lim_{n \to \infty} \left| \frac{z}{n+1} \right| =[\text{For any fixed complex number z, |z| is just a constant}] |z| \cdot \lim_{n \to \infty} \frac{1}{n+1} = |z| \cdot 0 = 0$. Since L = 0 < 1 regardless of what complex number $z$ we choose, the convergence condition is always satisfied. Hence, the series converges for every $z \in \mathbb{C}$ and its radius of convergence is $R = \infty$. We call such functions entire.

Functions defined by power series with an infinite radius of convergence are called entire functions. They are analytic everywhere in the complex plane. Polynomials, sines, cosines, and exponentials are all entire.

Differentiate term by term

A fundamental theorem of complex analysis guarantees that a power series can be differentiated term by term strictly inside its radius of convergence. This is a deep and beautiful theorem in complex analysis: inside the disk of convergence, a power series defines a holomorphic function, and holomorphic functions are infinitely differentiable with derivatives obtained by differentiating the series termwise.

Since $R = \infty$, we can differentiate our function anywhere in $\mathbb{C}$:

f’(z) = $\sum_{n=1}^{\infty} \frac{nz^{n-1}}{n!} = \sum_{n=1}^{\infty} \frac{z^{n-1}}{(n-1)!} =[\text{Readjusting the index}] \sum_{n=0}^{\infty} a_nz^n$ = f(z).

Why does the sum start at $n=1$? For n = 0, the term is $\frac{d}{dz}(\frac{z^0}{0!}) = \frac{d}{dz}(1) = 0$. The derivative of this constant is $0$ (By convention, 0! = 1 and in power-series $0^0 = 1$).

Furthermore, evaluating the original series at $z=0$ gives: f(0) = $\sum_{n=0}^{\infty} \frac{0^n}{n!} = \frac{0^0}{0!} + \frac{0^1}{1!} + \frac{0^2}{2!} + \dots = 1$.

We have just proven that our power series is the unique solution to the initial value problem (IVP): $f'(z) = f(z), \quad \text{with } f(0) = 1.$

The Existence and Uniqueness Theorem for ODEs guarantees that this IVP has a unique solution. But we already know a function that satisfies this IVP: the exponential $e^z$. Therefore, the power series f(z) must be that function $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$.

Local vs global power-series representations

A key question arises: is every analytic function (a function f is analytic at a point a if it is differentiable at a and in a small disk around a) equal to a power series or representable as a power series on its domain?

The answer is nuanced. A single power series rarely describes f on its entire domain; its radius is limited by the nearest singularity. $g(z) = \frac{1}{1-z}$ is well-defined and differentiable everywhere in the complex plane except at the single point z = 1 where it blows up, so its domain of analyticity is It can be expressed as a power series in a disk of convergence: $\frac{1}{1-z} = \sum_{n=0}^{\infty} z^n, \forall |z| \lt 1$.
Examples: $\frac{1}{1-z}$ centered at $0$, the nearest singularity is at $z=1$, so $R=1$.
For $\frac{1}{1+z^2}$ centered at $0$, the singularities are at $z=i$ and $z=-i$, both distance $1$ from $0$, so again $R=1$.
For $e^z$, there are no singularities in the finite complex plane, so $R=\infty$.

It can be expressed as a power series centered at z = 0 in a disk of convergence: $\frac{1}{1-z} = \sum_{n=0}^{\infty} z^n, \forall |z| \lt 1$. In other words, the series correctly represents the function only within the open disk $|z| < 1$.

The function g(z) is perfectly analytic at z = 2 (since $2 \neq 1$). If we plug $z = 2$ into the function, we get a perfectly valid output: $g(2) = \frac{1}{1-2} = -1$. However, its power series $\sum z^n$ diverges there ($\sum 2^n = 1 + 2 + 4 + 8 + \dots$ wildly diverges).

This is not a contradiction! It just means that a single power series centered at a = 0 is not a global representation of the function, but a local representation. The series grows outward from the origin until it hits the singularity at $z=1$. It cannot cross or skip over the singularity to evaluate the function at $z=2$. The singularity acts as an impenetrable wall for the power series.

Taylor’s Theorem

If global power series representations fail, what guarantees do we have? Is every analytic function equal to some power series locally?, i.e. in some B(a; r) for r > 0 where a is a point of analyticity for f. The answer is locally, yes (Taylor’s Theorem).

Not all analytic functions can be represented by a single power series over their entire domain. However, every analytic function can be approximated by a power series in a small neighborhood around each point of analyticity. Every analytic function can be represented by a power series locally.

Taylor’s Theorem. If f is analytic on an open disk B(a; R) (a disk of radius R centered at a), then f(z) can be represented exactly by a unique power series within that entire disk: $f(z) = \sum_{n=0}^{\infty}a_n (z - a)^n, \forall z \in B(a; R)$

The implication is profound. It states that if a function is differentiable just once in a disk, it must also be infinitely differentiable and representable by a power series (it is perfectly equal to its Taylor series).

Furthermore, the theorem provides exact formulas for the unique coefficients $a_n$. They can be calculated in two equivalent ways:

1. Using Derivatives:
   $a_n = \frac{f^{(n)}(a)}{n!}$
   This is the familiar calculus formula.
2. Using the Cauchy Integral Formula:
   $a_n = \frac{1}{2\pi i}\oint_{C_r} \frac{f(w)}{(w-a)^{n+1}}dw$
   where $C_r$ is any circle of radius $r < R$ centered at $a$, oriented counterclockwise.
   $C_r=\{ \, a+re^{i\theta }:0\leq \theta <2\pi \, \}$
   This formula is astonishing: the coefficient of $(z-a)^n$ is encoded in the values of f on any small circle around a.
   This is the geometric heart of complex analysis.

The Bridge between Integrals and Series

How do these two formulas for $a_n$ connect? If a function $f$ is analytic, the Cauchy Integral Formula for Derivatives states that the $n$-th derivative at a point $a$ can be computed via a contour integral: $f^{(n)}(a) = \frac{n!}{2\pi i} \oint_{C_r} \frac{f(w)}{(w-a)^{n+1}} dw$

Dividing both sides by $n!$ perfectly bridges the gap between the calculus definition of Taylor coefficients and the topological/integral geometry of the complex plane.

Why this matters

Taylor’s Theorem in complex analysis is not just a computational tool —it is a structural theorem about the nature of holomorphic functions:
✔ Holomorphic functions are determined by their values on any small curve ($C_r=\{ \, a+re^{i\theta }:0\leq \theta <2\pi \, \}$ is the boundary of a smaller disk entirely contained in the region where f is analytic and the values of f on this one small circle determine every derivative of f at the center.) The entire Taylor series of an analytic function is encoded in its values on any arbitrarily small circle around the center.
✔ Holomorphic functions are infinitely differentiable
✔ Holomorphic functions equal their Taylor series
✔ Holomorphic functions are very rigid: knowing them on a tiny set determines them everywhere in a domain