Learning Outcomes
- Determine if a series converges absolutely or conditionally.
Section 16.8 Absolute Convergence (SQ8)
Subsection 16.8.1 Activities
Activity 16.113.
Recall the series \(\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n}\) from Activity 16.104.
(a)
Does the series \(\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n}\) converge or diverge?
(b)
Does the series \(\displaystyle \sum_{n=1}^\infty \left|\frac{(-1)^n}{n}\right|\) converge or diverge?
Activity 16.114.
Consider the series \(\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n^2}\text{.}\)
(a)
Does the series \(\displaystyle \sum_{n=1}^\infty \frac{(-1)^n}{n^2}\) converge or diverge?
(b)
Does the series \(\displaystyle \sum_{n=1}^\infty \left|\frac{(-1)^n}{n^2}\right|\) converge or diverge?
Definition 16.115.
Given a series
\begin{equation*}
\sum a_n
\end{equation*}
we say that \(\displaystyle \sum a_n\) is absolutely convergent if \(\displaystyle \sum |a_n|\) converges.
Activity 16.116.
Consider the series: \(\displaystyle \sum_{n=1}^\infty \frac{(-1)^nn!}{(2n)!}\text{.}\)
(a)
Does the series \(\displaystyle \sum_{n=1}^\infty \frac{(-1)^nn!}{(2n)!}\) converge or diverge? (Recall Fact 16.105.)
(b)
Compute \(|a_n|\text{.}\)
(c)
Does the series \(\displaystyle \sum_{n=1}^\infty \frac{(-1)^nn!}{(2n)!}\) converge absolutely?
Fact 16.117.
Notice that Fact 16.105 and Fact 16.106 both involve taking absolute values to determine convergence. As such, series that are convergent by either the Ratio Test or the Root Test are also absolutely convergent (by applying the same test after taking the absolute value).
Activity 16.118.
Consider the series: \(\displaystyle \sum_{n=1}^\infty -n\text{.}\)
(a)
Does the series \(\displaystyle \sum_{n=1}^\infty -n\) converge or diverge?
(b)
Compute \(|a_n|\text{.}\)
(c)
Does the series \(\displaystyle \sum_{n=1}^\infty -n\) converge absolutely?
Activity 16.119.
For each of the following series, determine if the series is convergent, and if the series is absolutely convergent.
(a)
\(\displaystyle \sum_{n=1}^\infty \frac{n^2(-1)^n}{n^3+1}\text{.}\)(b)
\(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\text{.}\)(c)
\(\displaystyle \sum_{n=1}^\infty (-1)^n \left(\frac{2}{3}\right)^n\text{.}\)Activity 16.120.
If you know a series \(\displaystyle \sum a_n\) is absolutely convergent, what can you conclude about whether or not \(\displaystyle \sum a_n\) is convergent?
- We cannot determine if \(\displaystyle \sum a_n\) is convergent.
- \(\displaystyle \sum a_n\) is convergent since it “grows slower” than \(\displaystyle \sum |a_n|\) (and falls slower than \(\displaystyle \sum -|a_n|\)).
Fact 16.121.
If \(\displaystyle \sum a_n\) is absolutely convergent, then it must be convergent.
Activity 16.122.
Find 3 series that are convergent but not absolutely convergent (recall Fact 16.78, Section 16.6).
Subsection 16.8.2 Videos
Subsection 16.8.3 Exercises
Exercises available at
https://tbil.org/preview/calculus/exercises/#/bank/SQ8/
