TBIL-Cal Series Convergence Strategy (SQ9)

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Section 8.9 Series Convergence Strategy (SQ9)

Learning Outcomes

Identify appropriate convergence tests for various series.

Subsection 8.9.1 Activities

Activity 8.9.1.

Which test for convergence is the best first test to apply to any series

\sum_{k = 1}^{\infty} a_{k} ?

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

\sum a_{k} ?

When we find a convergent series $\sum b_{k}$ where $0 \leq a_{k} \leq b_{k}$
When we find a divergent series $\sum b_{k}$ where $0 \leq a_{k} \leq b_{k}$
When we find a convergent series $\sum b_{k}$ where $0 \leq b_{k} \leq a_{k}$
When we find a divergent series $\sum b_{k}$ where $0 \leq b_{k} \leq a_{k}$

Activity 8.9.3.

Which test(s) for convergence would we use for a series

\sum a_{k}

where

a_{k}

involves

k^{t h}

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

\sum a r^{k} ?

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

\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

\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

\sum (- 1)^{k} a_{k} ?

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 $a_{n} \to 0,$ $\sum a_{n}$ diverges
Geometric Series: $\sum a r^{k}$ converges if $- 1 < r < 1$ and diverges otherwise
$p$ -series: $\sum \frac{1}{n^{p}}$ 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

\sum_{k = 3}^{\infty} \frac{2}{\sqrt{k - 2}} .

(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

\sum_{k = 1}^{\infty} \frac{k}{1 + 2 k} .

(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

\sum_{k = 0}^{\infty} \frac{2 k^{2} + 1}{k^{3} + k + 1} .

(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

\sum_{k = 0}^{\infty} \frac{100^{k}}{k!} .

(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

\sum_{k = 1}^{\infty} \frac{2^{k}}{5^{k}} .

(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

\sum_{k = 1}^{\infty} \frac{k^{3} - 1}{k^{5} + 1} .

(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

\sum_{k = 2}^{\infty} \frac{3^{k - 1}}{7^{k}} .

(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

\sum_{k = 2}^{\infty} \frac{1}{k^{k}} .

(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

\sum_{k = 1}^{\infty} \frac{(- 1)^{k + 1}}{\sqrt{k + 1}} .

(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

\sum_{k = 2}^{\infty} \frac{1}{k \ln (k)} .

(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)

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

(b)

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

Subsection 8.9.2 Videos

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

Subsection 8.9.3 Exercises

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