Learning Outcomes
- Use the divergence, alternating series, and integral tests to determine if a series converges or diverges.
Section 16.5 Basic Convergence Tests (SQ5)
Subsection 16.5.1 Activities
Activity 16.64.
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 16.65.
If the series \(\displaystyle\sum a_n\) is convergent, then \(\displaystyle\lim_{n\rightarrow\infty} a_n=0\text{.}\)
Fact 16.66. 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 16.67.
Which of the series from Activity 16.64 diverge by Fact 16.66?
Fact 16.68.
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 16.69.
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 16.70.
In Activity 16.69, 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 16.71.
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 16.72. 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 16.73.
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 16.74. 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 16.75.
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 16.72?
Activity 16.76.
Prove Fact 16.74.
Activity 16.77.
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 16.78. 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 16.79.
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 16.80.
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 16.81. 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 16.82.
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.
Subsection 16.5.2 Videos
Subsection 16.5.3 Exercises
Exercises available at
https://tbil.org/preview/calculus/exercises/#/bank/SQ5/
