Learning Outcomes
-
Use the direct comparison and limit comparison tests to determine if a series converges or diverges.
Section 8.6 Comparison Tests (SQ6)
Subsection 8.6.1 Activities
Activity 8.6.1.
Let \(\{a_n\}_{n=1}^\infty\) be a sequence, with infinite series \(\displaystyle \sum_{n=1}^\infty a_n=a_1+a_2+\cdots \text{.}\) Suppose \(\{b_n\}_{n=1}^\infty\) is a sequence where each \(b_n=3a_n\text{,}\) with infinite series \(\displaystyle \sum_{n=1}^\infty b_n=\sum_{n=1}^\infty 3a_n=3a_1+3a_2+\cdots \text{.}\)
(a)
If \(\displaystyle \sum_{n=1}^\infty a_n=5\) what can be said about \(\displaystyle\sum_{n=1}^\infty b_n\text{?}\)
-
\(\displaystyle\sum_{n=1}^\infty b_n\) converges but the value cannot be determined.
-
\(\displaystyle\sum_{n=1}^\infty b_n\) converges to \(3\cdot 5=15\text{.}\)
-
\(\displaystyle\sum_{n=1}^\infty b_n\) converges to some value other than 15.
-
\(\displaystyle\sum_{n=1}^\infty b_n\) diverges.
-
It cannot be determined whether \(\displaystyle\sum_{n=1}^\infty b_n\) converges or diverges.
(b)
If \(\displaystyle \sum_{n=1}^\infty a_n\) diverges, what can be said about \(\displaystyle\sum_{n=1}^\infty b_n\text{?}\)
-
\(\displaystyle\sum_{n=1}^\infty b_n\) converges but the value cannot be determined.
-
\(\displaystyle\sum_{n=1}^\infty b_n\) converges and the value can be determined.
-
\(\displaystyle\sum_{n=1}^\infty b_n\) diverges.
-
It cannot be determined whether \(\displaystyle\sum_{n=1}^\infty b_n\) converges or diverges.
Activity 8.6.3.
Using Fact 8.4.2, we know the geometric series
\begin{equation*}
\displaystyle \sum_{n=0}^\infty \frac{1}{2^n}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^n}+\cdots=\frac{1}{1-\frac{1}{2}}=2.
\end{equation*}
(a)
What can we say about the series
\begin{equation*}
\displaystyle 3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots+\frac{3}{2^n}+\cdots?
\end{equation*}
-
\(\displaystyle 3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots+\frac{3}{2^n}+\cdots\) converges to \(3\cdot 2=6\text{.}\)
-
\(\displaystyle 3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots+\frac{3}{2^n}+\cdots\) converges to some value other than 6.
-
\(\displaystyle 3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots+\frac{3}{2^n}+\cdots\) diverges.
(b)
What do you think we can say about the series
\begin{equation*}
\displaystyle \frac{3.1}{2}+\frac{3.01}{4}+\frac{3.001}{8}+\cdots+\frac{3+(0.1)^n}{2^n}+\cdots?
\end{equation*}
-
\(\displaystyle 3+\frac{3.1}{2}+\frac{3.01}{4}+\frac{3.001}{8}+\cdots+\frac{3+(0.1)^n}{2^n}+\cdots\) converges to \(3\cdot 2=6\text{.}\)
-
\(\displaystyle 3+\frac{3.1}{2}+\frac{3.01}{4}+\frac{3.001}{8}+\cdots+\frac{3+(0.1)^n}{2^n}+\cdots\) converges to some value other than 6.
-
\(\displaystyle 3+\frac{3.1}{2}+\frac{3.01}{4}+\frac{3.001}{8}+\cdots+\frac{3+(0.1)^n}{2^n}+\cdots\) diverges.
Activity 8.6.4.
From Fact 8.4.2, we know
\begin{equation*}
\displaystyle 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{n}+\cdots
\end{equation*}
diverges.
(a)
What can we say about the series
\begin{equation*}
\displaystyle 5+\frac{5}{2}+\frac{5}{3}+\frac{5}{4}+\cdots+\frac{5}{n}+\cdots?
\end{equation*}
-
\(\displaystyle 5+\frac{5}{2}+\frac{5}{3}+\frac{5}{4}+\cdots+\frac{5}{n}+\cdots\) converges to a known value we can compute.
-
\(\displaystyle 5+\frac{5}{2}+\frac{5}{3}+\frac{5}{4}+\cdots+\frac{5}{n}+\cdots\) converges to some unknown value.
-
\(\displaystyle 5+\frac{5}{2}+\frac{5}{3}+\frac{5}{4}+\cdots+\frac{5}{n}+\cdots\) diverges.
(b)
What do you think we can say about the series
\begin{equation*}
\displaystyle 4.9+\frac{4.99}{2}+\frac{4.999}{3}+\frac{4.9999}{4}+\cdots+\frac{5-(0.1)^n}{n}+\cdots?
\end{equation*}
-
\(\displaystyle 4.9+\frac{4.99}{2}+\frac{4.999}{3}+\frac{4.9999}{4}+\cdots+\frac{5-(0.1)^n}{n}+\cdots\) converges to a known value we can compute.
-
\(\displaystyle 4.9+\frac{4.99}{2}+\frac{4.999}{3}+\frac{4.9999}{4}+\cdots+\frac{5-(0.1)^n}{n}+\cdots\) converges to some unknown value.
-
\(\displaystyle 4.9+\frac{4.99}{2}+\frac{4.999}{3}+\frac{4.9999}{4}+\cdots+\frac{5-(0.1)^n}{n}+\cdots\) diverges.
Fact 8.6.5. The Limit Comparison Test.
Let \(\displaystyle\sum a_n\) and \(\displaystyle\sum b_n\) be series with positive terms. If
\begin{equation*}
\lim_{n \to \infty} \frac{b_n}{a_n} = c
\end{equation*}
for some positive (finite) constant \(c\text{,}\) then \(\displaystyle\sum a_n\) and \(\displaystyle\sum b_n\) either both converge or both diverge.
Activity 8.6.6.
Recall that
\begin{equation*}
\displaystyle \sum_{n=1}^\infty \frac{1}{2^n}
\end{equation*}
converges.
(a)
Let \(b_n=\frac{1}{n}\text{.}\) Compute \(\displaystyle \lim_{n\to\infty}\frac{\frac{1}{n}}{\frac{1}{2^n}}\text{.}\)
-
\(-\infty\text{.}\)
-
\(0\text{.}\)
-
\(\displaystyle \frac{1}{2}\text{.}\)
-
\(1\text{.}\)
-
\(\infty\text{.}\)
(b)
Does \(\displaystyle\sum_{n=1}^\infty \frac{1}{n}\) converge or diverge?
(c)
Let \(\displaystyle b_n=\frac{1}{n^2}\text{.}\) Compute \(\displaystyle \lim_{n\to\infty}\frac{\frac{1}{n^2}}{\frac{1}{2^n}}\text{.}\)
-
\(\infty\text{.}\)
-
\(\ln(2)\text{.}\)
-
\(1\text{.}\)
-
\(\displaystyle \frac{1}{2}\text{.}\)
-
\(-\infty\text{.}\)
(d)
Does \(\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}\) converge or diverge?
(e)
Let \(\displaystyle\sum a_n\) and \(\displaystyle\sum b_n\) be series with positive terms. If
\begin{equation*}
\lim_{n \to \infty} \frac{b_n}{a_n}
\end{equation*}
diverges, can we conclude that \(\displaystyle \sum b_n\) converges or diverges?
Activity 8.6.7.
We wish to determine if \(\displaystyle \sum_{n=1}^\infty \frac{1}{4^n-1}\) converges or diverges using Fact 8.6.5.
(a)
Compute
\begin{equation*}
\displaystyle\lim_{n\to\infty}\frac{\frac{1}{4^n-1}}{\frac{1}{4^n}}.
\end{equation*}
(b)
Does the geometric series \(\displaystyle \sum_{n=1}^\infty \frac{1}{4^n}\) converge or diverge by Fact 8.4.2?
(c)
Does \(\displaystyle \sum_{n=1}^\infty \frac{1}{4^n-1}\) converge or diverge?
Activity 8.6.8.
We wish to determine if \(\displaystyle \sum_{n=2}^\infty \frac{2}{\sqrt{n+3}}\) converges or diverges using Fact 8.6.5.
(a)
To which of the following should we compare \(\displaystyle \{a_n\}=\left\{\frac{2}{\sqrt{n+3}}\right\}\text{?}\)
-
\(\displaystyle \left\{\frac{1}{n}\right\}\text{.}\)
-
\(\displaystyle \left\{\frac{1}{\sqrt{n}}\right\}\text{.}\)
-
\(\displaystyle \left\{\frac{1}{n^2}\right\}\text{.}\)
-
\(\displaystyle \left\{\frac{1}{2^n}\right\}\text{.}\)
(b)
Compute \(\displaystyle \lim_{n\to\infty} \frac{b_n}{a_n}\text{.}\)
(c)
Compute \(\displaystyle \lim_{n\to\infty} \frac{a_n}{b_n}\text{.}\)
(d)
What is true about \(\displaystyle \lim_{n\to\infty} \frac{b_n}{a_n}\) and \(\displaystyle \lim_{n\to\infty} \frac{a_n}{b_n}\text{?}\)
-
Their values are reciprocals.
-
Their values negative reciprocals.
-
They are both positive finite constants.
-
Only one value is a finite positive constant.
-
One value is \(0\) and the other value is infinite.
(e)
Does the series \(\displaystyle \sum_{n=2}^\infty \frac{1}{\sqrt{n}}\) converge or diverge?
(f)
Using your chosen sequence and Fact 8.6.5, does \(\displaystyle \sum_{n=2}^\infty \frac{2}{\sqrt{n+3}}\) converge or diverge?
Activity 8.6.9.
We wish to determine if \(\displaystyle \sum_{n=1}^\infty \frac{3}{n^2+8n+5}\) converges or diverges using Fact 8.6.5.
(a)
To which of the following should we compare \(\displaystyle \{x_n\}=\left\{\frac{3}{n^2+8n+5} \right\}\text{?}\)
-
\(\displaystyle \left\{\frac{1}{n}\right\}\text{.}\)
-
\(\displaystyle \left\{\frac{1}{\sqrt{n}}\right\}\text{.}\)
-
\(\displaystyle \left\{\frac{1}{n^2}\right\}\text{.}\)
-
\(\displaystyle \left\{\frac{1}{2^n}\right\}\text{.}\)
(b)
Using your chosen sequence and Fact 8.6.5, does \(\displaystyle \frac{3}{n^2+8n+5}\) converge or diverge?
Activity 8.6.10.
Use Fact 8.6.5 to determine if the series \(\displaystyle \sum_{n=5}^\infty \frac{2}{4^n}\) converges or diverges.
Activity 8.6.11.
(a)
Suppose that \(\displaystyle \sum_{n=0}^\infty a_n\) converges. What could be said about \(\{b_n\}\text{?}\)
-
\(\displaystyle \sum_{n=0}^\infty b_n\) converges.
-
\(\displaystyle \sum_{n=0}^\infty b_n\) diverges.
-
Whether or not \(\displaystyle \sum_{n=0}^\infty b_n\) converges or diverges cannot be determined with this information.
(b)
Suppose that \(\displaystyle \sum_{n=1}^\infty a_n=\sum_{n=1}^\infty \frac{1}{n+1}\) which diverges. Which of the following statements are true?
-
\(\displaystyle 0\leq \frac{1}{2n^2} \leq \frac{1}{n+1}\) for each \(n \geq 1\) and \(\displaystyle \sum_{n=1}^\infty \frac{1}{2n^2}\) is a convergent \(p\)-series where \(p=2\text{.}\)
-
\(\displaystyle 0\leq \frac{1}{2n}\leq \frac{1}{n+1}\) for each \(n \geq 1\) and \(\displaystyle \sum_{n=1}^\infty \frac{1}{2n}\) is a divergent \(p\)-series where \(p=1\text{.}\)
(c)
Suppose that \(\displaystyle \sum_{n=0}^\infty a_n\) was some series that diverges. What could be said about \(\{b_n\}\text{?}\)
-
\(\displaystyle \sum_{n=0}^\infty b_n\) converges.
-
\(\displaystyle \sum_{n=0}^\infty b_n\) diverges.
-
Whether or not \(\displaystyle \sum_{n=0}^\infty b_n\) converges or diverges cannot be determined with this information.
(d)
Suppose that \(\displaystyle \sum_{n=0}^\infty b_n\) diverges. What could be said about \(\{a_n\}\text{?}\)
-
\(\displaystyle \sum_{n=0}^\infty a_n\) converges.
-
\(\displaystyle \sum_{n=0}^\infty a_n\) diverges.
-
Whether or not \(\displaystyle \sum_{n=0}^\infty a_n\) converges or diverges cannot be determined with this information.
(e)
Suppose that \(\displaystyle \sum_{n=0}^\infty b_n=\sum_{n=0}^\infty \frac{1}{3^n}\) which converges. Which of the following statements are true?
-
\(\displaystyle 0\leq \frac{1}{3^n} \leq \frac{1}{2^n}\) for each \(n\) and \(\displaystyle \sum_{n=0}^\infty \frac{1}{2^n}\) is a convergent geometric series where \(\displaystyle |r|=\frac{1}{2} \lt 1\text{.}\)
-
\(\displaystyle 0\leq \frac{1}{3^n} \leq 1\) for each \(n\) and \(\displaystyle \sum_{n=0}^\infty 1\) diverges by the Divergence Test.
(f)
Suppose that \(\displaystyle \sum_{n=0}^\infty b_n\) was some series that converges. What could be said about \(\{a_n\}\text{?}\)
-
\(\displaystyle \sum_{n=0}^\infty a_n\) converges.
-
\(\displaystyle \sum_{n=0}^\infty a_n\) diverges.
-
Whether or not \(\displaystyle \sum_{n=0}^\infty a_n\) converges or diverges cannot be determined with this information.
Fact 8.6.12.
Suppose we have sequences \(\{a_n\}, \{b_n\}\) so that for some \(k\) we have that \(0\leq b_n\leq a_n\) for each \(k\geq n\text{.}\) Then we have the following results:
-
If \(\displaystyle\sum_{k=n}^\infty a_n\) converges, then so does \(\displaystyle\sum_{k=n}^\infty b_n\text{.}\)
-
If \(\displaystyle\sum_{k=n}^\infty b_n\) diverges, then so does \(\displaystyle\sum_{k=n}^\infty a_n\text{.}\)
Activity 8.6.13.
Suppose that you were handed positive sequences \(\{a_n\}, \{b_n\}\text{.}\) For the first few values \(a_n\geq b_n\text{,}\) but after that what happens is unclear until \(n=100\text{.}\) Then for any \(n\geq 100\) we have that \(a_n \leq b_n\text{.}\)
Figure 182. Plots of \(\{a_n\}, \{b_n\}\)
Plots of sequences \(\{a_n\}, \{b_n\}\) where \(a_n\geq b_n\geq 0\) initially but eventually \(a_n\leq b_n\geq 0\text{.}\)
(a)
How might we best utilize Fact 8.6.12 to determine the convergence of \(\displaystyle \sum_{n=0}^\infty a_n\) or \(\displaystyle \sum_{n=0}^\infty b_n\text{?}\)
-
Since \(a_n\) is sometimes greater than, and sometimes less than \(b_n\text{,}\) there is no way to utilize Fact 8.6.12.
-
Since initially, we have \(b_n\leq a_n\text{,}\) we can utilize Fact 8.6.12 by assuming \(a_n\geq b_n\text{.}\)
-
Since we can rewrite \(\displaystyle \sum_{n=0}^\infty a_n=\sum_{n=0}^{99} a_n+\sum_{n=100}^\infty a_n\) and \(\displaystyle \sum_{n=0}^\infty b_n=\sum_{n=0}^{99} b_n+\sum_{n=100}^\infty b_n\) and \(\displaystyle \sum_{n=0}^{99} a_n, \sum_{n=0}^{99} b_n\) are necessarily finite, we can compare \(\displaystyle \sum_{n=100}^\infty a_n, \sum_{n=100}^\infty b_n\) with Fact 8.6.12.
Fact 8.6.14. The Direct Comparison Test.
Let \(\displaystyle\sum a_n\) and \(\displaystyle\sum b_n\) be series with positive terms. If there is a \(k\) such that \(b_n\leq a_n\) for each \(n\geq k\text{,}\) then:
-
If \(\displaystyle \sum a_n\) converges, then so does \(\displaystyle \sum b_n\text{.}\)
-
If \(\displaystyle \sum b_n\) diverges, then so does \(\displaystyle \sum a_n\text{.}\)
Activity 8.6.15.
Suppose we wish to determine if \(\displaystyle \sum_{n=1}^\infty \frac{1}{2n+3}\) converged using Fact 8.6.14.
(a)
Does \(\displaystyle \sum_{n=1}^\infty \frac{1}{3n}\) converge or diverge?
(b)
For which value \(k\) is \(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k\text{?}\)
-
\(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k=0\text{.}\)
-
\(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k=1\text{.}\)
-
\(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k=2\text{.}\)
-
\(\displaystyle\frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k=3\text{.}\)
-
There is no \(k\) for which \(\displaystyle \frac{1}{3n}\leq \frac{1}{2n+3}\) for each \(n\geq k\text{.}\)
(c)
Use Fact 8.6.14 and compare \(\displaystyle \sum_{n=1}^\infty \frac{1}{2n+3}\) to \(\displaystyle \sum_{n=1}^\infty \frac{1}{3n}\) to determine if \(\displaystyle \sum_{n=1}^\infty \frac{1}{2n+3}\) converges or diverges.
Activity 8.6.16.
Suppose we wish to determine if \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2+5}\) converged using Fact 8.6.14.
(a)
Which series should we compare \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2+5}\) to best utilize Fact 8.6.14?
-
\(\displaystyle\sum_{n=1}^\infty \frac{1}{n}\text{.}\)
-
\(\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}\text{.}\)
-
\(\displaystyle\sum_{n=1}^\infty \frac{1}{2^n}\text{.}\)
-
\(\displaystyle\sum_{n=1}^\infty \frac{1}{n+5}\text{.}\)
-
\(\displaystyle\sum_{n=1}^\infty \frac{1}{n^2+5}\text{.}\)
-
\(\displaystyle\sum_{n=1}^\infty \frac{1}{2^n+5}\text{.}\)
(b)
Using your chosen series and Fact 8.6.14, does \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2+5}\) converge or diverge?
Activity 8.6.17.
For each of the following series, determine if it converges or diverges, and explain your choice.
(a)
\(\displaystyle \sum_{n= 4 }^\infty \frac{3}{\log\left(n\right) + 2}.\)(b)
\(\displaystyle \sum_{n= 3 }^\infty \frac{1}{n^{2} + 2 \, n + 1}.\)Subsection 8.6.2 Videos
Subsection 8.6.3 Exercises
Exercises available at
https://library.tbil.org/2025/calculus/exercises/#/bank/SQ6/
