Section 6.1 Power Series and Functions
References.
- Calculus, Early Transcendentals by Stewart, Chapter 11, Sections 8 & 9.
We have already seen one important type of a power series: the geometric series
\begin{equation*}
\sum_{n=0}^\infty a x^n = a + ax + ax^2 + \cdots + ax^n + \cdots
\end{equation*}
and determined that
- it converges for some values of \(x\) (\(|x| < 1\text{,}\) so \(-1 < x < 1\)) but not others, and
- for \(x\) values giving convergence, the value is \(\ds\frac{a}{1-x}\text{.}\)
These two questions will arise for other power series:
- for which \(x\) values does the series converge, and
- when it does converge, what is the value of the sum? That is, what function does the power series give?
Definition 6.1.2. Power Series.
A Power Series is a series of the form
\begin{equation*}
\sum_{n=0}^\infty c_n x^n = c_0 + c_1 x + c_2 x^2 + \cdots + c_n x^n + \cdots
\end{equation*}
where \(x\) is some number, and the \(c_n\) are constants: that is, they do not depending on \(x\text{.}\)
More generally, powers of \(x-a\) can be used for some constant \(a\text{,}\) so the most general power series is of the form
\begin{equation*}
\sum_{n=0}^\infty c_n (x-a)^n = c_0 + c_1 (x-a) + c_2 (x-a)^2 + \cdots + c_n (x-a)^n + \cdots
\end{equation*}
The constant \(a\) is called the center of the series.
Example 6.1.3.
For which values of \(x\) does one get convergence of the series
\begin{equation*}
\sum_{n=1}^\infty \frac{x^n}{n} = x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots + \frac{x^n}{n} + \cdots
\end{equation*}
The sum converges for \(-1 \leq x < 1\) and diverges otherwise.
Example 6.1.4.
For which values of \(x\) does one get convergence of the series
\begin{equation*}
\sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3!} + \cdots
\end{equation*}
The sum converges for all \(x\text{;}\) that is \(|x| < \infty\text{.}\)
Example 6.1.5.
For which values of \(x\) does one get convergence of the series
\begin{equation*}
\sum_{n=0}^\infty n! x^n = 1 + x + 2x^2 + 6x^3 + \cdots
\end{equation*}
The sum converges only for \(x = 0\text{;}\) that is “\(|x| \leq 0\)”.
Two patterns are worth noting in the above examples:
- One primarily gets convergence for \(|x|\) “small enough”, divergence for sufficiently large \(|x|\text{.}\)
- Convergence and divergence is shown primarily by the Ratio Test (or the Root Test).
- Exceptions to the previous two observations occur at the two borderline points where \(|x|\) has the largest value for which convergence might occur.
- These two borderline \(x\) values give \(\rho = 1\) in the Ratio Test (or the Root Test), so those tests give no answer; thus to determine convergence we must use some other method, like the Alternating Series Test or one of the comparison tests.
The above patterns are in fact universal:
Theorem 6.1.6.
For a given power series \(\ds \sum_{n=0}^\infty c_n (x-a)^n\) one of the following is true
- There is a positive number \(R\) such that the series converges (absolutely) for \(|x-a| < R\text{,}\) and diverges for \(|x-a|>R\text{.}\)
- The series converges (absolutely) for all \(x\text{:}\) that is, for \(|x-a| <\infty\text{,}\) or \(x \in (\infty,\infty)\text{.}\) (Informally, “\(R=\infty\)”.)
- The series converges only for \(x=a\text{:}\) that is, for \(|x-a| \leq 0\text{,}\) or \(x \in [a,a]\text{.}\) (“\(R=0\)”).
The first case is silent on two values of \(x\text{:}\) \(a+R\) and \(a-R\text{.}\) At each of these, one can have either convergence or divergence: see for example the “50-50” case of Example 6.1.3 above.
The number \(R\) in case (i) is called the Radius of Convergence. In fact, we can make sense of a radius of convergence in every case:
- in case (ii), we say the radius of convergence is \(R=\infty\text{;}\)
- in the boring case (iii), we say the radius of convergence is \(R=0\text{.}\)
Also, in every case the \(x\) values giving convergence form an interval, which we call the Interval of Convergence:
- In case (i), the interval of convergence can be \((a-R,a+R)\text{,}\) \([a-R,a+R)\text{,}\) \((a-R,a+R]\text{,}\) or \([a-R,a+R]\text{.}\)
- In case (ii) it is \((-\infty,\infty)\)
- In case (iii) it is just \([a,a]\text{.}\)
Example 6.1.7.
Find the radius of convergence and interval of convergence of the series
\begin{equation*}
\sum_{n=0}^\infty (-2)^n \sqrt{n} (x-2)^n.
\end{equation*}
The root test shows that the sum converges for \(|x-2| < 1/2\) and diverges for \(|x-2| > 1/2\text{,}\) so the radius of convergence is \(R = 1/2\text{.}\) Then testing the end cases \(x = 3/2,5/3\text{,}\) where \(|x-2| = 1/2\text{,}\) gives divergence in each case, so the interval of convergence is \((3/2, 5/2)\text{.}\)
Example 6.1.8.
Find the radius of convergence and interval of convergence for the series
\begin{equation*}
\sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{2^{2n}(n!)^2}
\end{equation*}
The series for \(J_0(x)\) converges for all \(x\text{,}\) so the radius of convergence is \(R=\infty\text{,}\) and so with no extra work needed, the interval of convergence is \((-\infty,\infty)\text{.}\)
Study Guide.
Study Calculus Volume 2, Section 6.1 3 ; in particular
- The defintions of a Power Series, its Center and its Radius of Convergence.
- Theorem 1 about the possibilites for which \(x\) values give convergence.
- Examples 1, 2 and 3 (focus on “radius” more than “interval”)
- Checkpoints 1 and 3
- and one or several exercises from each of the following ranges: 1–4, 5 and 6, 13–16, 23–26, 29–32.
openstax.org/books/calculus-volume-2/pages/6-1-power-series-and-functions
en.wikipedia.org/wiki/Bessel_function
openstax.org/books/calculus-volume-2/pages/6-1-power-series-and-functions