Power Series¶

Definition¶

Let $a \in \mathbb{R}\text{.}$ A power series centred at $a$ is expressed as $\displaystyle\sum_{n=0}^\infty c_n (x-a)^n\text{,}$ where $(c_n)_{n=0}^\infty$ is a sequence of real numbers, independent of $x\text{.}$ We refer to $c_n$ as the $n$ -th coefficient of the power series.

Viewed as an Infinite Series of Functions, its $n$ th summand is $x\mapsto c_n(x-a)^n$ $a)^{n}$ . Convergence asks whether the partial sums of these functions have a limit at the chosen $x$ .

Interpretation¶

A power series is an infinite sum of power functions $c_n(x-a)^n$ $a)^{n}$ , a polynomial of unlimited degree centred at $a$ . For each fixed $x$ it is a numerical series that may or may not converge; the definition itself does not say for which $x$ it does.

The function 1/(1 - x) with partial sums of its geometric series with 2, 4 and 8 terms; they match it on the shaded interval from minus 1 to 1 and move away outside it.

The figure takes $a=0$ and $c_n=1$ for all $n$ , the geometric series $\displaystyle \sum_n x^n$ , and shows its partial sums with $2$ , $4$ and $8$ terms next to $1/(1-x)$ $x)$ . On the shaded interval $(-1,1)$ the partial sums approach $1/(1-x)$ $x)$ ; outside it the terms $x^n$ do not tend to $0$ , and the partial sums move away. The function $1/(1-x)$ exists for $x<-1$ as well, but the series does not represent it there.

Raise $N$ : inside the shaded interval of convergence the partial sums settle on the sum; outside they run away.

The centre $a$ and the coefficients $c_n$ determine the series. At $x=a$ every term with $n\ge1$ vanishes and the sum is $c_0$ ; here $(x-a)^0$ is read as $1$ even at $x=a$ $a$ . Away from the centre, convergence depends on the distance $|x-a|$ $a ∣$ , which is measured by the radius of convergence. Within that radius the sum is a differentiable function that can be differentiated term by term, by the theorem on convergence of power series.

Power series are how many standard functions are defined or computed. The series $1+x+\tfrac{x^2}{2}+\tfrac{x^3}{6}+\cdots$ in the Examples has sum $\mathrm{e}^x$ , the exponential function. A function that equals a power series near each point of an open interval is real analytic; the series is then built from the derivatives of the function at the centre, which returns later as the Taylor series.

Key Results¶

The definition is formal: it does not say for which $x$ the series converges. Every power series has a radius of convergence $R\in[0,+\infty]$ $[0, + \infty]$ : it converges absolutely for $|x-a|<R$ and diverges for $|x-a|>R\text{.}$

Examples¶

Any polynomial is a special case of a power series, since the coefficients $c_n$ are allowed to be zero.
$1+x+\dfrac{x^2}{2}+\dfrac{x^3}{6}+\cdots$ is a power series.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 2

Symbols: element of, infinity, real numbers, sum

Last updated: 20 October 2026 03:02