Learning Outcomes
- Determine the interval of convergence for a given power series.
Section 17.2 Convergence of Power Series (PS2)
Subsection 17.2.1 Activities
Activity 17.7.
Consider the series \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n\) where \(x\) is a real number.
(a)
If \(x=2\text{,}\) then \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n=\sum_{n=0}^\infty \frac{2^n}{n!}\text{.}\) What can be said about this series?
- The techniques we have learned so far allow us to conclude that \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n=\sum_{n=0}^\infty \frac{2^n}{n!}\) converges.
- The techniques we have learned so far allow us to conclude that \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n=\sum_{n=0}^\infty \frac{2^n}{n!}\) diverges.
- None of the techniques we have learned so far allow us to conclude whether \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n=\sum_{n=0}^\infty \frac{2^n}{n!}\) converges or diverges.
(b)
If \(x=-100\text{,}\) then \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n=\sum_{n=0}^\infty \frac{(-100)^n}{n!}\text{.}\) What can be said about this series?
- The techniques we have learned so far allow us to conclude that \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n=\sum_{n=0}^\infty \frac{(-100)^n}{n!}\) converges.
- The techniques we have learned so far allow us to conclude that \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n=\sum_{n=0}^\infty \frac{(-100)^n}{n!}\) diverges.
- None of the techniques we have learned so far allow us to conclude whether \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n=\sum_{n=0}^\infty \frac{(-100)^n}{n!}\) converges or diverges.
(c)
Suppose that \(x\) were some arbitrary real number. What can be said about this series?
- The techniques we have learned so far allow us to conclude that \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n\) converges.
- The techniques we have learned so far allow us to conclude that \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n\) diverges.
- None of the techniques we have learned so far allow us to conclude whether \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n\) converges or diverges.
Remark 17.8.
Consider a power series \(\displaystyle\sum a_n(x-c)^n\text{.}\) Recall from Fact 16.105 that if
\begin{align*}
\displaystyle \lim_{n\to \infty} \left|\frac{a_{n+1}(x-c)^{n+1}}{a_n(x-c)^n}\right| & < 1
\end{align*}
then \(\displaystyle\sum a_n(x-c)^n\) converges.
Then recall:
\begin{align*}
\displaystyle \lim_{n\to \infty} \left|\frac{a_{n+1}(x-c)^{n+1}}{a_n(x-c)^n}\right| & = \displaystyle \lim_{n\to \infty} \left|\frac{a_{n+1}(x-c)}{a_n}\right|\\
&=\displaystyle \lim_{n\to \infty} |x-c|\left|\frac{a_{n+1}}{a_n}\right| \\
&=\displaystyle |x-a|\lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right|.
\end{align*}
Activity 17.9.
Consider \(\displaystyle\sum_{n=0}^\infty \frac{1}{n^2+1}x^n.\)
(a)
Letting \(a_n=\frac{1}{n^2+1}\text{,}\) find \(\displaystyle \lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right|\text{.}\)
(b)
For what values of \(x\) is \(\displaystyle |x|\lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1\text{?}\)
- \(x < 1\text{.}\)
- \(0\leq x < 1\text{.}\)
- \(-1 < x < 1\text{.}\)
(c)
If \(x=1\text{,}\) does \(\displaystyle\sum_{n=0}^\infty \frac{1}{n^2+1}x^n\) converge?
(d)
If \(x=-1\text{,}\) does \(\displaystyle\sum_{n=0}^\infty \frac{1}{n^2+1}x^n\) converge?
(e)
Which of the following describe the values of \(x\) for which \(\displaystyle\sum_{n=0}^\infty \frac{1}{n^2+1}x^n\) converges?
- \((-1,1)\text{.}\)
- \([-1,1)\text{.}\)
- \((-1,1]\text{.}\)
- \([-1,1]\text{.}\)
Activity 17.10.
Consider \(\displaystyle\sum_{n=0}^\infty \frac{2^n}{5^n}(x-2)^n.\)
(a)
Letting \(a_n=\frac{2^n}{5^n}\text{,}\) find \(\displaystyle \lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right|\text{.}\)
(b)
For what values of \(x\) is \(\displaystyle |x-2|\lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1\text{?}\)
- \(-\frac{2}{5} < x < \frac{2}{5}\text{.}\)
- \(\frac{8}{5} < x < \frac{12}{5}\text{.}\)
- \(-\frac{5}{2} < x < \frac{5}{2}\text{.}\)
- \(-\frac{1}{2} < x < \frac{9}{2}\text{.}\)
(c)
If \(x=\frac{9}{2}\text{,}\) does \(\displaystyle\sum_{n=0}^\infty \frac{2^n}{5^n}(x-2)^n\) converge?
(d)
If \(x=-\frac{1}{2}\text{,}\) does \(\displaystyle\sum_{n=0}^\infty \frac{2^n}{5^n}(x-2)^n\) converge?
(e)
Which of the following describe the values of \(x\) for which \(\displaystyle\sum_{n=0}^\infty \frac{2^n}{5^n}(x-2)^n\) converges?
- \((-\frac{1}{2},\frac{9}{2})\text{.}\)
- \([-\frac{1}{2},\frac{9}{2})\text{.}\)
- \((-\frac{1}{2},\frac{9}{2}]\text{.}\)
- \([-\frac{1}{2},\frac{9}{2}]\text{.}\)
Activity 17.11.
Consider \(\displaystyle\sum_{n=0}^\infty \frac{n^2}{n!}\left(x+\frac{1}{2}\right)^n.\)
(a)
Letting \(c_n=\frac{n^2}{n!}\text{,}\) find \(\displaystyle \lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right|\text{.}\)
(b)
For what values of \(x\) is \(\displaystyle \left|x+\frac{1}{2}\right|\lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right| < 1\text{?}\)
- \(0\leq x < \infty\text{.}\)
- All real numbers.
(c)
What describes the values of \(x\) for which \(\displaystyle\sum_{n=0}^\infty \frac{n^2}{n!}\left(x+\frac{1}{2}\right)^n\) converges?
Fact 17.12.
Given the power series \(\displaystyle\sum a_n(x-c)^n\text{,}\) the center of convergence is \(x=a\text{.}\) The radius of convergence is
\begin{equation*}
r=\frac{1}{\displaystyle\lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right|}.
\end{equation*}
If \(\displaystyle\lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right|=0\text{,}\) we say that \(r=\infty\text{.}\)
The interval of convergence represents all possible values of \(x\) for which \(\displaystyle\sum a_n(x-c)^n\) converges, which is of the form:
- \(\displaystyle (c-r, c+r)\)
- \(\displaystyle [c-r, c+r)\)
- \(\displaystyle (c-r, c+r]\)
- \(\displaystyle [c-r, c+r]\)
Depending on if \(\displaystyle\sum a_n(x-c)^n\) converges when \(x=c-r\) or \(x=c+r\text{.}\)
If \(r=\infty\text{,}\) the interval of convergence is all real numbers.
Activity 17.13.
Find the center of convergence, radius of convergence, and interval of convergence for the series:
\begin{equation*}
\sum_{n=0}^\infty \frac{3^{n} \left(-1\right)^{n} {\left(x - 1\right)}^{n}}{n!}.
\end{equation*}
Activity 17.14.
Find the center of convergence, radius of convergence, and interval of convergence for the series:
\begin{equation*}
\sum_{n=0}^\infty \frac{3^{n} {\left(x + 2\right)}^{n}}{n}.
\end{equation*}
Activity 17.15.
Consider the power series \(\displaystyle \sum_{n=0}^\infty \frac{2^n+1}{n3^n}\left(x+1\right)^n\text{.}\)
(a)
What is the center of convergence for this power series?
(b)
What is the radius of convergence for this power series?
(c)
What is the interval of convergence for this power series?
(d)
If \(x=-0.5\text{,}\) does this series converge? (Use the interval of convergence.)
(e)
If \(x=1\text{,}\) does this series converge? (Use the interval of convergence.)
Subsection 17.2.2 Videos
Subsection 17.2.3 Exercises
Exercises available at
https://tbil.org/preview/calculus/exercises/#/bank/PS2/
