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Calculus for Team-Based Inquiry Learning: 2024 Edition

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Section 9.2 Convergence of Power Series (PS2)

Subsection 9.2.1 Activities

Activity 9.2.1.

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?
  1. 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.
  2. 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.
  3. 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?
  1. 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.
  2. 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.
  3. 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?
  1. The techniques we have learned so far allow us to conclude that \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n\) converges.
  2. The techniques we have learned so far allow us to conclude that \(\displaystyle \sum_{n=0}^\infty \frac{1}{n!}x^n\) diverges.
  3. 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 9.2.2.

Consider a power series \(\displaystyle\sum c_n(x-a)^n\text{.}\) Recall from Fact 8.7.6 that if
\begin{align*} \displaystyle \lim_{n\to \infty} \left|\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}\right| & < 1 \end{align*}
then \(\displaystyle\sum c_n(x-a)^n\) converges.
Then recall:
\begin{align*} \displaystyle \lim_{n\to \infty} \left|\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}\right| & = \displaystyle \lim_{n\to \infty} \left|\frac{c_{n+1}(x-a)}{c_n}\right|\\ &=\displaystyle \lim_{n\to \infty} |x-a|\left|\frac{c_{n+1}}{c_n}\right| \\ &=\displaystyle |x-a|\lim_{n\to \infty} \left|\frac{c_{n+1}}{c_n}\right|. \end{align*}

Activity 9.2.3.

Consider \(\displaystyle\sum_{n=0}^\infty \frac{1}{n^2+1}x^n.\)
(a)
Letting \(c_n=\frac{1}{n^2+1}\text{,}\) find \(\displaystyle \lim_{n\to \infty} \left|\frac{c_{n+1}}{c_n}\right|\text{.}\)
(b)
For what values of \(x\) is \(\displaystyle |x|\lim_{n\to \infty} \left|\frac{c_{n+1}}{c_n}\right| < 1\text{?}\)
  1. \(x < 1\text{.}\)
  2. \(0\leq x < 1\text{.}\)
  3. \(-1 < x < 1\text{.}\)
(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. \((-1,1)\text{.}\)
  2. \([-1,1)\text{.}\)
  3. \((-1,1]\text{.}\)
  4. \([-1,1]\text{.}\)

Activity 9.2.4.

Consider \(\displaystyle\sum_{n=0}^\infty \frac{2^n}{5^n}(x-2)^n.\)
(a)
Letting \(c_n=\frac{2^n}{5^n}\text{,}\) find \(\displaystyle \lim_{n\to \infty} \left|\frac{c_{n+1}}{c_n}\right|\text{.}\)
(b)
For what values of \(x\) is \(\displaystyle |x-2|\lim_{n\to \infty} \left|\frac{c_{n+1}}{c_n}\right| < 1\text{?}\)
  1. \(-\frac{2}{5} < x < \frac{2}{5}\text{.}\)
  2. \(\frac{8}{5} < x < \frac{12}{5}\text{.}\)
  3. \(-\frac{5}{2} < x < \frac{5}{2}\text{.}\)
  4. \(-\frac{1}{2} < x < \frac{9}{2}\text{.}\)
(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?
  1. \((-\frac{1}{2},\frac{9}{2})\text{.}\)
  2. \([-\frac{1}{2},\frac{9}{2})\text{.}\)
  3. \((-\frac{1}{2},\frac{9}{2}]\text{.}\)
  4. \([-\frac{1}{2},\frac{9}{2}]\text{.}\)

Activity 9.2.5.

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 \(\displaystyle \lim_{n\to \infty} \left|\frac{c_{n+1}}{c_n}\right|\text{.}\)
(b)
For what values of \(x\) is \(\displaystyle \left|x+\frac{1}{2}\right|\lim_{n\to \infty} \left|\frac{c_{n+1}}{c_n}\right| < 1\text{?}\)
  1. \(0\leq x < \infty\text{.}\)
  2. 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?

Activity 9.2.7.

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 9.2.8.

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 9.2.9.

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 9.2.2 Videos

Figure 189. Video: Determine the interval of convergence for a given power series

Subsection 9.2.3 Exercises

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