TBIL-Cal Basic Convergence Tests (SQ5)

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.
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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.

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\(\displaystyle \sum_{n=0}^\infty n^2\text{.}\)

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\(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{n+1}{n}\text{.}\)

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\(\displaystyle \sum_{n=0}^\infty (-1)^n\text{.}\)

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\(\displaystyle \sum_{n=1}^\infty \frac{1}{n}\text{.}\)

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\(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\text{.}\)

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Fact 8.5.2.

If the series \(\displaystyle\sum a_n\) is convergent, then \(\displaystyle\lim_{n\rightarrow\infty} a_n=0\text{.}\)

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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.

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Activity 8.5.4.

Which of the series from Activity 8.5.1 diverge by Fact 8.5.3?

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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.

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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.

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(a)

Determine which of the following inequalities hold(s).

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\(\displaystyle\frac{1}{3}+\frac{1}{4}\lt \frac{1}{2}\text{.}\)

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\(\displaystyle\frac{1}{3}+\frac{1}{4}\gt \frac{1}{2}\text{.}\)

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\(S_4\geq S_2+\displaystyle\frac{1}{2}\text{.}\)

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\(S_4\leq S_2+\displaystyle\frac{1}{2}\text{.}\)

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\(S_4= S_2+\displaystyle\frac{1}{2}\text{.}\)

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(b)

Determine which of the following inequalities hold(s).

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\(\displaystyle\frac{1}{2}\lt \frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\text{.}\)

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\(\displaystyle\frac{1}{2}\gt \frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\text{.}\)

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\(S_8=S_4+\displaystyle\frac{1}{2}\text{.}\)

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\(S_8\geq S_4+\displaystyle\frac{1}{2}\text{.}\)

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\(S_8\leq S_4+\displaystyle\frac{1}{2}\text{.}\)

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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?

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The harmonic series converges.
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The harmonic series diverges.
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Activity 8.5.8.

Consider the series \(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{1}{n^2}\text{.}\)

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(a)

We want to compare this series to an improper integral. Which of the following is the best candidate?

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\(\displaystyle\int_1^\infty x^2 \, dx\text{.}\)

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\(\displaystyle\int_1^\infty \displaystyle\frac{1}{x^3} \, dx\text{.}\)

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\(\displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)

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\(\displaystyle\int_1^\infty \displaystyle\frac{1}{x} \, dx\text{.}\)

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\(\displaystyle\int_1^\infty x \, dx\text{.}\)

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(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 \(\Delta x=1\text{.}\)

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The sum \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using right Riemann sums where \(\Delta x=1\text{.}\)

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The sum \(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using left Riemann sums where \(\Delta x=1\text{.}\)

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The sum \(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2}\) corresponds to approximating the integral chosen above using right Riemann sums where \(\Delta x=1\text{.}\)

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(c)

Using the Riemann sum interpretation of the series, identify which of the following inequalities holds.

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\(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} \leq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)

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\(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2} \geq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)

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\(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2} \geq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)

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\(\displaystyle \sum_{n=2}^\infty \frac{1}{n^2} \leq \displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{.}\)

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(d)

What can we say about the improper integral \(\displaystyle\int_1^\infty \displaystyle\frac{1}{x^2} \, dx\text{?}\)

This improper integral converges.

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This improper integral diverges.

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(e)

What do you think is true about the series \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\text{?}\)

The series converges.

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The series diverges.

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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.

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Activity 8.5.10.

Consider the \(p\)-series \(\displaystyle \sum_{n=1}^\infty \frac{1}{n^p}\text{.}\)

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(a)

Recall that the harmonic series diverges. What value of \(p\) corresponds to the harmonic series?

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\(p=-1\text{.}\)

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\(p=1\text{.}\)

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\(p=-2\text{.}\)

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\(p=2\text{.}\)

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\(p=0\text{.}\)

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(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.

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This series converges.

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This series diverges.

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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.

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Activity 8.5.12.

Consider the series \(\displaystyle \sum_{n=1}^\infty \displaystyle\frac{1}{n^2+1}\text{.}\)

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(a)

If we aim to use the integral test, what is an appropriate choice for \(f(x)\text{?}\)

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\(\displaystyle \frac{1}{x^2}\text{.}\)

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\(x^2+1\text{.}\)

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\(\displaystyle \frac{1}{x^2+1}\text{.}\)

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\(x^2\text{.}\)

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\(\displaystyle \frac{1}{x}\text{.}\)

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(b)

Does the series converge or diverge by Fact 8.5.9?

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Activity 8.5.13.

Prove Fact 8.5.11.

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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.

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\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) converges.

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\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) converges.

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\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) diverges.

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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,

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there is an integer \(N\) such that \(u_n\geq u_{n+1}\) for all \(n\geq N\text{,}\) and

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\(\displaystyle\lim_{n\rightarrow\infty}u_n=0\text{.}\)

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Activity 8.5.16.

What conclusions can you now make?

\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) diverges.

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\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n}\) converges.

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\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) converges.

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\(\displaystyle \sum_{n=1}^\infty (-1)^n \frac{1}{n^2}\) diverges.

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Activity 8.5.17.

For each of the following series, use the Divergence, Alternating Summation or Integral test to determine if the series converges.

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(a)

\(\displaystyle \sum_{n=1}^\infty \frac{2 \, {\left(n^{2} + 2\right)}}{n^{2}}.\)

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(b)

\(\displaystyle \sum_{n=1}^\infty \frac{1}{n^{4}}.\)

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(c)

\(\displaystyle \sum_{n=1}^\infty \frac{3 \, \left(-1\right)^{n}}{4 \, n}.\)

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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):