Learning Outcomes
- Use the divergence, alternating series, and integral tests to determine if a series converges or diverges.
Section 8.5 Basic Convergence Tests (SQ5)
Subsection 8.5.1 Activities
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 improper 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)
- 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.
