Learning Outcomes
- Identify appropriate convergence tests for various series.
Section 8.9 Series Convergence Strategy (SQ9)
Subsection 8.9.1 Activities
Activity 8.9.1.
Which test for convergence is the best first test to apply to any series \(\displaystyle \sum_{k=1}^\infty a_k\text{?}\)
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
Activity 8.9.2.
In which of the following scenarios can we successfully apply the Direct Comparison Test to determine the convergence of the series \(\displaystyle \sum a_k\text{?}\)
- When we find a convergent series \(\displaystyle \sum b_k\) where \(0\leq a_k\leq b_k\)
- When we find a divergent series \(\displaystyle \sum b_k\) where \(0\leq a_k\leq b_k\)
- When we find a convergent series \(\displaystyle \sum b_k\) where \(0\leq b_k\leq a_k\)
- When we find a divergent series \(\displaystyle \sum b_k\) where \(0\leq b_k\leq a_k\)
Activity 8.9.3.
Which test(s) for convergence would we use for a series \(\displaystyle \sum a_k\) where \(a_k\) involves \(k^{th}\) powers?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
Activity 8.9.4.
Which test(s) for convergence would we use for a series of the form \(\displaystyle \sum ar^k\text{?}\)
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
Activity 8.9.5.
Which test(s) for convergence would we use for a series \(\displaystyle \sum a_k\) where \(a_k\) involves factorials and powers?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
Activity 8.9.6.
Which test(s) for convergence would we use for a series \(\displaystyle \sum a_k\) where \(a_k\) is a rational function?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
Activity 8.9.7.
Which test(s) for convergence would we use for a series of the form \(\displaystyle \sum (-1)^ka_k\text{?}\)
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
Fact 8.9.8.
Here is a strategy checklist when dealing with series:
- The divergence test: unless \(a_n\rightarrow 0\text{,}\) \(\displaystyle \sum a_n\) diverges
- Geometric Series: \(\displaystyle \sum ar^k\) converges if \(-1<r<1\) and diverges otherwise
- \(p\)-series: \(\displaystyle \sum \frac{1}{n^p}\) converges if \(p>1\) and diverges otherwise
- Series with no negative terms: try the ratio test, root test, integral test, or try to compare to a known series with the comparison test or limit comparison test
- Series with some negative terms: check for absolute convergence
- Alternating series: use the alternating series test (Leibniz’s Theorem)
- Anything else: consider the sequence of partial sums, possibly rewriting the series in a different form, hope for the best
Activity 8.9.9.
Consider the series \(\displaystyle \sum_{k=3}^\infty \frac{2}{\sqrt{k-2}}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.10.
Consider the series \(\displaystyle \sum_{k=1}^\infty \frac{k}{1+2k}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.11.
Consider the series \(\displaystyle \sum_{k=0}^\infty \frac{2k^2+1}{k^3+k+1}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.12.
Consider the series \(\displaystyle \sum_{k=0}^\infty \frac{100^k}{k!}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.13.
Consider the series \(\displaystyle \sum_{k=1}^\infty \frac{2^k}{5^k}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.14.
Consider the series \(\displaystyle \sum_{k=1}^\infty \frac{k^3-1}{k^5+1}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.15.
Consider the series \(\displaystyle \sum_{k=2}^\infty \frac{3^{k-1}}{7^k}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.16.
Consider the series \(\displaystyle \sum_{k=2}^\infty \frac{1}{k^k}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.17.
Consider the series \(\displaystyle \sum_{k=1}^\infty \frac{(-1)^{k+1}}{\sqrt{k+1}}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.18.
Consider the series \(\displaystyle \sum_{k=2}^\infty \frac{1}{k\ln(k)}\text{.}\)
(a)
Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?
- Divergence Test
- Geometric Series
- Integral Test
- Direct Comparison Test
- Limit Comparison Test
- Ratio Test
- Root Test
- Alternating Series Test
(b)
Apply an appropriate test to determine the convergence of this series.
- This series is convergent.
- This series is divergent.
Activity 8.9.19.
Determine which of the following series is convergent and which is divergent. Justify both choices with an appropriate test.
(a)
\(\displaystyle \sum_{n=1}^\infty \frac{4 \, \left(-1\right)^{n + 1} n^{2}}{2 \, n^{3} + 4 \, n^{2} + 5}.\)(b)
\(\displaystyle \sum_{n=1}^\infty \frac{n!}{3 \cdot 3^{n} n^{4}}.\)Subsection 8.9.2 Videos
Subsection 8.9.3 Exercises
Exercises available at
https://tbil.org/calculus/preview/exercises/#/bank/SQ9/
