A power series (centered at the origin) is an expression
of the form: $$\displaystyle \sum_{n=0}^{\infty} c_n x^n =
c_0+c_1x+c_2x^2+c_3x^3+\ldots$$ where $x$
is a variable and each
$c_n$ is a cofficient, which is a constant (perhaps
different constants, depending upon $n$). For example,
$$\displaystyle \sum_{n=0}^\infty \frac{x^n}{n!},\quad\text{ or
}\quad\sum_{n=0}^\infty \frac{x^n}{n+1}.$$In the first example
above, $c_n=\frac{1}{n!}$ since our terms are
$c_nx^n=\frac{1}{n!}x^n=\frac{x^n}{n!}$ Notice that since
$x$ is a variable, we could plug in $x=1$, say, and get the series
$\sum \frac{1}{n!}$ or plug in $x=\frac{1}{2}$ and get the series
$\sum \frac{1}{2^n n!}$. DO:
What is $c_n$ for the second example? What series would we
get if we let $x=1$? or $x=2$? Notice that
the $x$ is the same in every term, while the coefficient may vary
depending on $n$.
DO: Write, in summation
notation as well as the first few terms, the power series with
$c_n=1$ for all $n$. Does this series converge at
$x=\frac{3}{4}$? How about at $x={1.2}$?
DO: Write, in summation
notation as well as the first few terms, the power series
beginning at $n=1$, with $c_n=\frac{1}{n^2}$ for each $n$.
Does this series converge at $x={1}$? How about at $x=\frac{5}{4}$? Hint: remember your
convergence tests!
We are interested in which values of $x$ will give us a
convergent series. Such $x$ are like the domain of the power
series, since the series makes sense at values of $x$ for which it
converges, and makes no sense at other values of $x$ for which it
is divergent. It turns out that the $x$-values for which a power series converge
consist of an interval, with the center of the interval being the center of the
power series. (Here, as mentioned above but
not explained, we are looking at series centered
at the origin.) We will discuss this in more
detail as we proceed.
| Definitions
(for a power series, centered at the origin) The set of $x$ where the series converges is called the interval of convergence, and is an interval from $-R$ to $R$, where $R\ge0$ is a called the radius of convergence. |
The interval of convergence may include one, both, or no endpoints. Special cases will be when $R=0$, where the series converges at the single point $x=0$, and $R=\infty$, where the series converges on the entire real line. In notation, the possible intervals of convergence for a series centered at the origin are $$[-R,R],\quad [-R,R), \quad(-R,R], \quad(-R,R),\quad\{0\}=[0,0],\quad\text{ and }\quad(-\infty,\infty).$$