TBIL-Calc Absolute Convergence (SQ8)

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Section 8.8 Absolute Convergence (SQ8)

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Learning Outcomes

Determine if a series converges absolutely or conditionally.

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Activity 8.8.1.

Recall the series $\sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{n}$ from Activity 8.7.5.

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(a)

Does the series $\sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{n}$ converge or diverge?

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(b)

Does the series $\sum_{n = 1}^{\infty} | \frac{(- 1)^{n}}{n} |$ converge or diverge?

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Activity 8.8.2.

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

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(a)

Does the series $\sum_{n = 1}^{\infty} \frac{(- 1)^{n}}{n^{2}}$ converge or diverge?

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(b)

Does the series $\sum_{n = 1}^{\infty} | \frac{(- 1)^{n}}{n^{2}} |$ converge or diverge?

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Definition 8.8.3.

Given a series

\sum a_{n}

we say that $\sum a_{n}$ is absolutely convergent if $\sum | a_{n} |$ converges.

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Activity 8.8.4.

Consider the series: $\sum_{n = 1}^{\infty} \frac{(- 1)^{n} n!}{(2 n)!} .$

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(a)

Does the series $\sum_{n = 1}^{\infty} \frac{(- 1)^{n} n!}{(2 n)!}$ converge or diverge? (Recall Fact 8.7.6.)

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(b)

Compute $| a_{n} | .$

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(c)

Does the series $\sum_{n = 1}^{\infty} \frac{(- 1)^{n} n!}{(2 n)!}$ converge absolutely?

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Fact 8.8.5.

Notice that Fact 8.7.6 and Fact 8.7.7 both involve taking absolute values to determine convergence. As such, series that are convergent by either the Ratio Test or the Root Test are also absolutely convergent (by applying the same test after taking the absolute value).

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Activity 8.8.6.

Consider the series: $\sum_{n = 1}^{\infty} - n .$

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(a)

Does the series $\sum_{n = 1}^{\infty} - n$ converge or diverge?

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(b)

Compute $| a_{n} | .$

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(c)

Does the series $\sum_{n = 1}^{\infty} - n$ converge absolutely?

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Activity 8.8.7.

For each of the following series, determine if the series is convergent, and if the series is absolutely convergent.

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(a)

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

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(b)

\sum_{n = 1}^{\infty} \frac{1}{n^{2}}

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(c)

\sum_{n = 1}^{\infty} (- 1)^{n} {(\frac{2}{3})}^{n}

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Activity 8.8.8.

If you know a series $\sum a_{n}$ is absolutely convergent, what can you conclude about whether or not $\sum a_{n}$ is convergent?

We cannot determine if $\sum a_{n}$ is convergent.
$\sum a_{n}$ is convergent since it “grows slower” than $\sum | a_{n} |$ (and falls slower than $\sum - | a_{n} |$ ).

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Fact 8.8.9.

If $\sum a_{n}$ is absolutely convergent, then it must be convergent.

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Activity 8.8.10.

Find 3 series that are convergent but not absolutely convergent (recall Fact 8.5.15, Section 8.6).

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Subsection 8.8.1 Videos

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Figure 158. Video: Determine if a series converges absolutely or conditionally