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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.
  1. \(\displaystyle \sum_{n=0}^\infty n^2\text{.}\)
  2. \(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{n+1}{n}\text{.}\)
  3. \(\displaystyle \sum_{n=0}^\infty (-1)^n\text{.}\)
  4. \(\displaystyle \sum_{n=1}^\infty \frac{1}{n}\text{.}\)
  5. \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\text{.}\)

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).
  1. \(\displaystyle\frac{1}{3}+\frac{1}{4}\lt \frac{1}{2}\text{.}\)
  2. \(\displaystyle\frac{1}{3}+\frac{1}{4}\gt \frac{1}{2}\text{.}\)
  3. \(S_4\geq S_2+\displaystyle\frac{1}{2}\text{.}\)
  4. \(S_4\leq S_2+\displaystyle\frac{1}{2}\text{.}\)
  5. \(S_4= S_2+\displaystyle\frac{1}{2}\text{.}\)
(b)
Determine which of the following inequalities hold(s).
  1. \(\displaystyle\frac{1}{2}\lt \frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\text{.}\)
  2. \(\displaystyle\frac{1}{2}\gt \frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\text{.}\)
  3. \(S_8=S_4+\displaystyle\frac{1}{2}\text{.}\)
  4. \(S_8\geq S_4+\displaystyle\frac{1}{2}\text{.}\)
  5. \(S_8\leq S_4+\displaystyle\frac{1}{2}\text{.}\)

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?
  1. The harmonic series converges.
  2. 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?
  1. \(\displaystyle\int_1^\infty x^2 \, dx\text{.}\)
  2. \(\displaystyle\int_1^\infty \displaystyle\frac{1}{x^3} \, dx\text{.}\)
  3. \(\displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
  4. \(\displaystyle\int_1^\infty \displaystyle\frac{1}{x} \, dx\text{.}\)
  5. \(\displaystyle\int_1^\infty x \, dx\text{.}\)
(b)
Select the true statements below.
  1. The sum \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using left Riemann sums where \(\Delta x=1\text{.}\)
  2. The sum \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using right Riemann sums where \(\Delta x=1\text{.}\)
  3. The sum \(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using left Riemann sums where \(\Delta x=1\text{.}\)
  4. 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.
  1. \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} \leq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
  2. \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} \geq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
  3. \(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2} \geq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
  4. \(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2} \leq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)
(d)
What can we say about the improper integral \(\displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{?}\)
  1. This improper integral converges.
  2. This improper integral diverges.
(e)
What do you think is true about the series \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\text{?}\)
  1. The series converges.
  2. The series diverges.

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?
  1. \(p=-1\text{.}\)
  2. \(p=1\text{.}\)
  3. \(p=-2\text{.}\)
  4. \(p=2\text{.}\)
  5. \(p=0\text{.}\)
(b)
From Fact 8.5.9, what can we conclude about the \(p\)-series with \(p=2\text{?}\)
  1. There is not enough information to draw a conclusion.
  2. This series converges.
  3. This series diverges.

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 \(f(x)\text{?}\)
  1. \(\displaystyle \frac{1}{x^2}\text{.}\)
  2. \(x^2+1\text{.}\)
  3. \(\displaystyle \frac{1}{x^2+1}\text{.}\)
  4. \(x^2\text{.}\)
  5. \(\displaystyle \frac{1}{x}\text{.}\)
(b)
Does the series converge or diverge by Fact 8.5.9?

Activity 8.5.14.

Which of the following statements seem(s) most likely to be true?
  1. \(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) diverges.
  2. \(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) converges.
  3. \(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) converges.
  4. \(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) diverges.

Activity 8.5.16.

What conclusions can you now make?
  1. \(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) diverges.
  2. \(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) converges.
  3. \(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) converges.
  4. \(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) 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)
\(\displaystyle \sum_{n=1}^\infty \frac{3 \, \left(-1\right)^{n}}{4 \, n}.\)

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.

Subsection 8.5.2 Videos

Figure 180. Video: Use the divergence, alternating series, and integral tests to determine if a series converges or diverges

Subsection 8.5.3 Exercises