Section 8.5 Basic Convergence Tests (SQ5)
Learning Outcomes
Use the divergence, alternating series, and integral tests to determine if a series converges or diverges.
Activity 8.5.1.
Which of the following series seem(s) to diverge? It might be helpful to write out the first several terms.
\(\displaystyle \sum_{n=0}^\infty n^2\text{.}\)
\(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{n+1}{n}\text{.}\)
\(\displaystyle \sum_{n=0}^\infty (-1)^n\text{.}\)
\(\displaystyle \sum_{n=1}^\infty \frac{1}{n}\text{.}\)
\(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\text{.}\)
Fact 8.5.2.
If the series \(\displaystyle\sum a_n\) is convergent, then \(\displaystyle\lim_{n\rightarrow\infty} a_n=0\text{.}\)
Fact 8.5.3. The Divergence (\(n^{th}\) term) Test.
If the \(\displaystyle\lim_{n\rightarrow\infty} a_n\neq 0\text{,}\) then \(\displaystyle\sum a_n\) diverges.
Activity 8.5.4.
Which of the series from Activity 8.5.1 diverge by Fact 8.5.3?
Fact 8.5.5.
If \(a_n>0\) for all \(n\text{,}\) then \(\displaystyle\sum a_n\) is convergent if and only if the sequence of partial sums is bounded from above.
Activity 8.5.6.
Consider the so-called harmonic series, \(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{1}{n}\text{,}\) and let \(S_n\) be its \(n^{th}\) partial sum.
(a)
Determine which of the following inequalities hold(s).
\(\displaystyle\frac{1}{3}+\frac{1}{4}\lt \frac{1}{2}\text{.}\)
\(\displaystyle\frac{1}{3}+\frac{1}{4}\gt \frac{1}{2}\text{.}\)
\(S_4\geq S_2+\displaystyle\frac{1}{2}\text{.}\)
\(S_4\leq S_2+\displaystyle\frac{1}{2}\text{.}\)
\(S_4= S_2+\displaystyle\frac{1}{2}\text{.}\)
(b)
Determine which of the following inequalities hold(s).
\(\displaystyle\frac{1}{2}\lt \frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\text{.}\)
\(\displaystyle\frac{1}{2}\gt \frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\text{.}\)
\(S_8=S_4+\displaystyle\frac{1}{2}\text{.}\)
\(S_8\geq S_4+\displaystyle\frac{1}{2}\text{.}\)
\(S_8\leq S_4+\displaystyle\frac{1}{2}\text{.}\)
Activity 8.5.7.
In Activity 8.5.6, we found that \(S_4\geq S_2+\displaystyle\frac{1}{2}\) and \(S_8\geq S_4+\displaystyle\frac{1}{2}\text{.}\) Based on these inequalities, which statement seems most likely to hold?
The harmonic series converges.
The harmonic series diverges.
Activity 8.5.8.
Consider the series \(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{1}{n^2}\text{.}\)
(a)
We want to compare this series to an improper integral. Which of the following is the best candidate?
\(\displaystyle\int_1^\infty x^2 \, dx\text{.}\)
\(\displaystyle\int_1^\infty \displaystyle\frac{1}{x^3} \, dx\text{.}\)
\(\displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
\(\displaystyle\int_1^\infty \displaystyle\frac{1}{x} \, dx\text{.}\)
\(\displaystyle\int_1^\infty x \, dx\text{.}\)
(b)
Select the true statements below.
The sum \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using left Riemann sums where \(\Delta x=1\text{.}\)
The sum \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using right Riemann sums where \(\Delta x=1\text{.}\)
The sum \(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using left Riemann sums where \(\Delta x=1\text{.}\)
The sum \(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using right Riemann sums where \(\Delta x=1\text{.}\)
(c)
Using the Riemann sum interpretation of the series, identify which of the following inequalities holds.
\(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} \leq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
\(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} \geq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
\(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2} \geq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
\(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2} \leq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
(d)
What can we say about the iproper integral \(\displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{?}\)
This improper integral converges.
This improper integral diverges.
(e)
What do you think is true about the series \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\text{?}\)
The series converges.
The series diverges.
Fact 8.5.9. The Integral Test.
Let \(\{a_n\}\) be a sequence of positive numbers. If \(f(x)\) is continuous, positive, and decreasing, and there is some positive integer \(N\) such that \(f(n)=a_n\) for all \(n\geq N\text{,}\) then \(\displaystyle \sum_{n=N}^\infty a_n\) and \(\displaystyle\int_N^\infty \displaystyle f(x) \, dx\) both converge or both diverge.
Activity 8.5.10.
Consider the \(p\)-series \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^p}\text{.}\)
(a)
Recall that the harmonic series diverges. What value of \(p\) corresponds to the harmonic series?
\(p=-1\text{.}\)
\(p=1\text{.}\)
\(p=-2\text{.}\)
\(p=2\text{.}\)
\(p=0\text{.}\)
(b)
From Fact 8.5.9, what can we conclude about the \(p\)-series with \(p=2\text{?}\)
There is not enough information to draw a conclusion.
This series converges.
This series diverges.
Fact 8.5.11. The \(p\)-Test.
The series \(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{1}{n^p}\) converges for \(p\gt 1\text{,}\) and diverges otherwise.
Activity 8.5.12.
Consider the series \(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{1}{n^2+1}\text{.}\)
(a)
If we aim to use the integral test, what is an appropriate choice for \(f(x)\text{?}\)
\(\displaystyle \frac{1}{x^2}\text{.}\)
\(x^2+1\text{.}\)
\(\displaystyle \frac{1}{x^2+1}\text{.}\)
\(x^2\text{.}\)
\(\displaystyle \frac{1}{x}\text{.}\)
(b)
Does the series converge or diverge by Fact 8.5.9?
Activity 8.5.13.
Prove Fact 8.5.11.
Activity 8.5.14.
Which of the following statements seem(s) most likely to be true?
\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) diverges.
\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) converges.
\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) converges.
\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) diverges.
Fact 8.5.15. The Alternating Series Test (Leibniz's Theorem).
The series \(\displaystyle\sum (-1)^{n+1}u_n\) converges if all of the following conditions are satisfied:
\(u_n\) is always positive,
there is an integer \(N\) such that \(u_n\geq u_{n+1}\) for all \(n\geq N\text{,}\) and
\(\displaystyle\lim_{n\rightarrow\infty}u_n=0\text{.}\)
Activity 8.5.16.
What conclusions can you now make?
\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) diverges.
\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) converges.
\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) converges.
\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) diverges.
Activity 8.5.17.
For each of the following series, use the Divergence, Alternating Summation or Integral test to determine if the series converges.
(a)
\(\displaystyle \sum_{n=1}^\infty \frac{2 \, {\left(n^{2} + 2\right)}}{n^{2}}.\)(b)
\(\displaystyle \sum_{n=1}^\infty \frac{1}{n^{4}}.\)(c)
\(\displaystyle \sum_{n=1}^\infty \frac{3 \, \left(-1\right)^{n}}{4 \, n}.\)Fact 8.5.18. The Alternating Series Estimation Theorem.
If the alternating series \(\displaystyle\sum a_n=\displaystyle\sum (-1)^{n+1}u_n\) converges to \(L\) and has \(n^{th}\) partial sum \(S_n\text{,}\) then for \(n\geq N\) (as in the alternating series test):
\(|L-S_n|\) is less than \(|a_{n+1}|\text{,}\) and
\((L-S_n)\) has the same sign as \(a_{n+1}\text{.}\)
Activity 8.5.19.
Consider the so-called alternating harmonic series, \(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{(-1)^{n+1}}{n}\text{.}\)
(a)
Use the alternating series test to determine if the series converges.
(b)
If so, estimate the series using the first 3 terms.