TBIL-Cal Series Convergence Strategy (SQ9)

Section 8.9 Series Convergence Strategy (SQ9)

Learning Outcomes

Identify appropriate convergence tests for various series.
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Subsection 8.9.1 Activities

Activity 8.9.1.

Which test for convergence is the best first test to apply to any series \(\displaystyle \sum_{k=1}^\infty a_k\text{?}\)

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Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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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 \(\displaystyle \sum a_k\text{?}\)

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When we find a convergent series \(\displaystyle \sum b_k\) where \(0\leq a_k\leq b_k\)
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When we find a divergent series \(\displaystyle \sum b_k\) where \(0\leq a_k\leq b_k\)
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When we find a convergent series \(\displaystyle \sum b_k\) where \(0\leq b_k\leq a_k\)
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When we find a divergent series \(\displaystyle \sum b_k\) where \(0\leq b_k\leq a_k\)
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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?

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Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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Activity 8.9.4.

Which test(s) for convergence would we use for a series of the form \(\displaystyle \sum ar^k\text{?}\)

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Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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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?

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Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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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?

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Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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Activity 8.9.7.

Which test(s) for convergence would we use for a series of the form \(\displaystyle \sum (-1)^ka_k\text{?}\)

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Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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Fact 8.9.8.

Here is a strategy checklist when dealing with series:

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The divergence test: unless \(a_n\rightarrow 0\text{,}\) \(\displaystyle \sum a_n\) diverges
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Geometric Series: \(\displaystyle \sum ar^k\) converges if \(-1<r<1\) and diverges otherwise
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\(p\)-series: \(\displaystyle \sum \frac{1}{n^p}\) converges if \(p>1\) and diverges otherwise
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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
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Series with some negative terms: check for absolute convergence
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Alternating series: use the alternating series test (Leibniz’s Theorem)
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Anything else: consider the sequence of partial sums, possibly rewriting the series in a different form, hope for the best
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Activity 8.9.9.

Consider the series \(\displaystyle \sum_{k=3}^\infty \frac{2}{\sqrt{k-2}}\text{.}\)

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.10.

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

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.11.

Consider the series \(\displaystyle \sum_{k=0}^\infty \frac{2k^2+1}{k^3+k+1}\text{.}\)

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.12.

Consider the series \(\displaystyle \sum_{k=0}^\infty \frac{100^k}{k!}\text{.}\)

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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Geometric Series
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Integral Test
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Direct Comparison Test
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Limit Comparison Test
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Ratio Test
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Root Test
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Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.13.

Consider the series \(\displaystyle \sum_{k=1}^\infty \frac{2^k}{5^k}\text{.}\)

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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Geometric Series
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🔗
Integral Test
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🔗
Direct Comparison Test
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🔗
Limit Comparison Test
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🔗
Ratio Test
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🔗
Root Test
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🔗
Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.14.

Consider the series \(\displaystyle \sum_{k=1}^\infty \frac{k^3-1}{k^5+1}\text{.}\)

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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🔗
Geometric Series
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🔗
Integral Test
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🔗
Direct Comparison Test
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🔗
Limit Comparison Test
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🔗
Ratio Test
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🔗
Root Test
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🔗
Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.15.

Consider the series \(\displaystyle \sum_{k=2}^\infty \frac{3^{k-1}}{7^k}\text{.}\)

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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Geometric Series
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🔗
Integral Test
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🔗
Direct Comparison Test
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🔗
Limit Comparison Test
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🔗
Ratio Test
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🔗
Root Test
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🔗
Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.16.

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

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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Geometric Series
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🔗
Integral Test
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🔗
Direct Comparison Test
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🔗
Limit Comparison Test
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🔗
Ratio Test
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🔗
Root Test
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🔗
Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.17.

Consider the series \(\displaystyle \sum_{k=1}^\infty \frac{(-1)^{k+1}}{\sqrt{k+1}}\text{.}\)

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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Geometric Series
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🔗
Integral Test
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🔗
Direct Comparison Test
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🔗
Limit Comparison Test
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🔗
Ratio Test
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🔗
Root Test
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🔗
Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.18.

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

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(a)

Which test(s) seem like the most appropriate one(s) to test for convergence or divergence?

Divergence Test
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🔗
Geometric Series
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🔗
Integral Test
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🔗
Direct Comparison Test
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🔗
Limit Comparison Test
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🔗
Ratio Test
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🔗
Root Test
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🔗
Alternating Series Test
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(b)

Apply an appropriate test to determine the convergence of this series.

This series is convergent.
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This series is divergent.
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Activity 8.9.19.

Determine which of the following series is convergent and which is divergent. Justify both choices with an appropriate test.

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(a)

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

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(b)

\(\displaystyle \sum_{n=1}^\infty \frac{n!}{3 \cdot 3^{n} n^{4}}.\)

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Subsection 8.9.2 Videos

Figure 186. Video: Identify appropriate convergence tests for various series

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Subsection 8.9.3 Exercises

Exercises available at https://library.tbil.org/2025/calculus/exercises/#/bank/SQ9/

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