Power Series

We will now look at an important type of series known as a power series which we define as follows.

Definition: A Power Series in Powers of $x - c$ is a series that takes on the form $\sum_{n=0}^{\infty} a_n(x - c)^n = a_0 + a_1(x - c) + a_2(x - c)^2 + a_3(x - c)^3 + ...$ where $a_0, a_1, a_2, a_3, ...$ are the Coefficients of The Power Series and $c$ is called the Center of Convergence of the power series.

We note that the power series is a function of a variable $x$, and that the power series will either converge or diverge for each value of $x$ chosen. If the series converges for some $x$, then the power series becomes a representation of some function for those values of $x$. If the series diverges for some $x$, then the power series is no longer a representation of a specific function for some $x$.

For example, consider the geometric series $\sum_{n=0}^{\infty} x^{n} = 1 + x + x^2 + ...$. Recall that if $\mid x \mid < 1$ then $\sum_{n=0}^{\infty} x^{n} = \frac{1}{1 - x}$. Therefore for $-1 < x < 1$, the series $\sum_{n=0}^{\infty} x^n$ is a power series representation of the function $f(x) = \frac{1}{1- x}$ on the interval $(-1, 1)$. We note that since all geometric series diverge if $\mid x \mid ≥ 1$, then the power series $\sum_{n=0}^{\infty} x^n$ is not a representation of $f$ on the interval $(-\infty, -1] \cup [1, \infty)$. The diagram below illustrates how this power series represents the function on the interval $(-1, 1)$ where the blue curve if the power series and the green curve if the function $f$:

Now we note that the geometric series $\sum_{n=0}^{\infty} x^n$ has a center of convergence at $c = 0$. Notice that the interval to which this series convergences is for $(-1, 1) = \{ x \in \mathbb{R} : -1 < x < 1 \}$ and that $c = 0$ is indeed the center of this interval (hence the name center of convergence).

Now the following theorem will outline what sort of intervals of convergence we can expect from power series.

Theorem 1: Let $\sum_{n=0}^{\infty} a_n(x - c)^n$ be a power series. Then one of the following conclusions can be made regarding the points of convergence for the power series: 1) The series converges only at the center of convergence $c$. 2) The series converges for every $x \in \mathbb{R}$. 3) There exists a positive number $R \in \mathbb{R}$ such that the series converges if $\mid x - c \mid < R$ and diverges if $\mid x - c \mid > R$ where the converge and diverges of $x$ such that $\mid x - c \mid = R$ is undetermined.

• Proof: We note that every power series converges at it's center of convergence, that is at the value $x = c$ since $\sum_{n=0}^{\infty} a_n(x - c)^n \biggr \rvert_{x=c} = \sum_{n=0}^{\infty} a_n(c-c)^n = \sum_{n=0}^{\infty} 0$, and this sum converges to $0$, specifically, this convergence is absolute convergence. We have thus shown that the validity of the first possibility.

• We now need to show that if a series converges at any $x_0 \neq c$ then it converges absolutely to every $x$ that is even closer to the center of convergence $c$, that is $\mid x - c \mid < \mid x_0 - c \mid$. So the convergence of any $x_0 \neq c$ will imply the absolute convergence for every $x$ such that $c - x_0 < x < c + x_0$. Now suppose that the power series $\sum_{n=0}^{\infty} a_n(x_0 - c)^n$ is convergent. We know by the divergence theorem that then $\lim_{n \to \infty} a_n(x_0 - c)^n = 0$. Therefore the sequence $\{ a_n(x_0 - c)^n \}$ converges and is bounded so $\forall n \in \mathbb{N}$ there exists a positive real number $K$ such that $\biggr \rvert a_n(x_0 - c)^n \biggr \rvert < K$.

• Now suppose that $r = \frac{\mid x - c \mid}{\mid x_0 - c \mid} < 1$. This happens for all $x$ such that $\lvert x - c \rvert < \lvert x_0 - c \rvert$ (the distance between $x$ and $c$ is less than the distance between $x_0$ and $c$), and so:

(1)

\begin{align} \quad \quad \sum_{n=0}^{\infty} \biggr \rvert a_n (x - c)^n \biggr \rvert = \sum_{n=0}^{\infty} \biggr \rvert a_n \biggr \rvert \biggr \rvert (x - c)^n \biggr \rvert = \sum_{n=0}^{\infty} \biggr \rvert a_n \biggr \rvert \biggr \rvert (x_0 - c)^n \biggr \rvert \biggr \rvert \frac{x - c}{x_0 - c} \biggr \rvert^n < K \sum_{n=0}^{\infty} \biggr \rvert \frac{x - c}{x_0 - c} \biggr \rvert^n = K \sum_{n=0}^{\infty} r^n = \frac{K}{1 - r} < \infty \end{align}

• Therefore the power series $\sum_{n=0}^{\infty} a_n(x_0 - c)^n$ is absolutely convergent which satisfies the second and third possibilities. $\blacksquare$

Definition: The Interval of Convergence $R$ of a power series $\sum_{n=0}^{\infty} a_n(x - c)^n$ is the interval of $x$ values centered at the the center of convergence $c$ for which the power series converges.

From the theorem above, we note that there are only 6 possibilities for the center of convergence summarized by the following corollary.

Corollary 1: Let $\sum_{n=0}^{\infty} a_n(x - c)^n$ be a power series. If this power series only converges at $x = c$ then the interval of convergence is the closed interval $[c, c] = \{ c \}$. If the power series converges for all $x \in \mathbb{R}$ then the interval of convergence is the entire real line $(-\infty, \infty) = \{ x \in \mathbb{R} \}$. Otherwise, the interval of convergence of a power series is some interval centered at the center of convergence $c$, that is either: 1) $[c - R, c + R] = \{ x \in \mathbb{R} : c - R ≤ x ≤ c + R \}$. 2) $(c - R, c + R) = \{ x \in \mathbb{R} : c - R < x < c + R \}$. 3) $[c - R, c + R) = \{ x \in \mathbb{R} : c - R ≤ x < c + R \}$. 4) $(c - R, c + R] = \{ x \in \mathbb{R} : c - R < x ≤ c + R \}$.

We note that on the interval $(c - R, c + R)$ the power series $\sum_{n=0}^{\infty} a_n(x - c)^n$ is absolutely convergent. We must check the end points of the power series when $x = c - R$ and $x = c + R$ to determine if the power series converges at $c - R$ and $c + R$.