For more information, please see full course syllabus of Calculus BC

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For more information, please see full course syllabus of Calculus BC

For more information, please see full course syllabus of Calculus BC

### Absolute Convergence

- If converges then “absolutely converges”
- Convergence check
- Test for absolute convergence
- If not absolutely convergent, test for conditional convergence or divergence

### Absolute Convergence

Does the series ∑

_{n = 1}^{∞}( − 1)^{n + 1}[1/(n^{2})] converge absolutely?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge - ∑
_{n = 1}^{∞}| ( − 1)^{n + 1}[1/(n^{2})] | = ∑_{n = 1}^{∞}| [1/(n^{2})] |

Apply P - series properties

∑

∑

_{n = 1}^{∞}| [1/(n^{2})] | = 0 Thus the series converges completelyDoes the series ∑

_{n = 1}^{∞}( − 1)^{n + 1}[(n − 1)/(n^{2}+ 2)] converge absolutely?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge - ∑
_{n = 1}^{∞}| ( − 1)^{n + 1}[(n − 1)/(n^{2}+ 2)] | = ∑_{n = 1}^{∞}| [(n − 1)/(n^{2}+ 2)] | - Apply Harmonic series properties

= [1/n] = diverges

Thus the series does not absolutely converge

Thus the series does not absolutely converge

Does the series ∑

_{n = 1}^{∞}[(( − 1)^{n + 1})/(√n + 3)] absolutely converge?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge - ∑
_{n = 1}^{∞}| [(( − 1)^{n + 1})/(√n + 3)] | = ∑_{n = 1}^{∞}| [1/(√n + 3)] | - Apply Comparison Test
- ∑
_{n = 1}^{∞}| [1/(√n + 3)] | compared with ∑_{n = 1}^{∞}| [1/(√n )] | - Apply P - Series properties
- p = [1/2] < 1 = diverges

Thus the series does not absolutely converge

Does the series ∑

_{n = 1}^{∞}[(( − 1)^{n})/(3^{n − 1})] absolutely converge?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge - ∑
_{n = 1}^{∞}| [(( − 1)^{n})/(3^{n − 1})] | = ∑_{n = 1}^{∞}| [1/(3^{n − 1})] |

a = 1r = [1/3]|r|< 1, thus the series absolutely converges

Does the series ∑

_{n = 1}^{∞}[(( − 1)^{n + 1})/(^{4}√{n + 2})] absolutely converge?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge - ∑
_{n = 1}^{∞}| [(( − 1)^{n + 1})/(^{4}√{n + 2})] | = ∑_{n = 1}^{∞}| [1/(^{4}√{n + 2})] |

∑

Thus the series does not absolutely converge

_{n = 1}^{∞}| [1/(^{4}√{n + 2})] | = ∑_{n = 1}^{∞}| [1/(( n + 2 )^{1/4})] | p = [1/4] < 1 = divergesThus the series does not absolutely converge

Does the series ∑

_{n = 1}^{∞}[(( − 1)^{n + 1}n^{2})/(^{3}√{n})] absolutely converge?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge

= ∑

_{n = 1}^{∞}| n^{5/3}| = ∞ Thus the series does not absolutely convergeDoes the series ∑

_{n = 1}^{∞}[(2( − 1)^{n}(n + 2)^{2})/(n^{2}+ 1)] absolutely converge?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge - ∑
_{n = 1}^{∞}| [(2( − 1)^{n}(n + 2)^{2})/(n^{2}+ 1)] | = ∑_{n = 1}^{∞}| [(2(n + 2)^{2})/(n^{2}+ 1)] | - = ∑
_{n = 1}^{∞}| [(2(n^{2}+ 2n + 4))/(n^{2}+ 1)] | - = ∑
_{n = 1}^{∞}| [(2n^{2}+ 4n + 8)/(n^{2}+ 1)] | - = [(2n
^{2})/(n^{2})] - = 2

Thus the series does absolutely converges

Does the series ∑

_{n = 1}^{∞}[(( − 1)^{n})/(3n + 1)] absolutely converge?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge - ∑
_{n = 1}^{∞}| [(( − 1)^{n})/(3n + 1)] | = ∑_{n = 1}^{∞}| [1/(3n + 1)] |

Let b

_{n}= [1/n]lim_{n∞}[([1/(3n + 1)])/([1/n])] = [1/3] Thus the series absolutely convergesDoes the series ∑

_{n = 1}^{∞}[(( − 1)^{n + 2}(n − 1))/(n^{3})] absolutely converge?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge - ∑
_{n = 1}^{∞}| [(( − 1)^{n + 2}(n − 1))/(n^{3})] | = ∑_{n = 1}^{∞}| [(n − 1)/(n^{3})] |

Let b

_{n}= [1/(n^{2})]lim_{n∞}[([(n − 1)/(n^{3})])/([1/(n^{2})])] = 1 Thus the series absolutely convergesDoes the series ∑

_{n = 1}^{∞}[(( − 1)^{n + 1}2^{n})/n!] absolutely converge?- Determine if ∑
_{n = 1}^{∞}| a_{n}| converge - ∑
_{n = 1}^{∞}| [(( − 1)^{n + 1}2^{n})/n!] | = ∑_{n = 1}^{∞}| [(2^{n})/n!] |

Apply the Ratio Test

∑

∑

_{n = 1}^{∞}| [(2^{n})/n!] | = 0 Thus the series does absolutely converges*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Absolute Convergence

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Absolute Convergence 0:12
- Example 1 0:52
- Example 2 3:42
- Example 3 5:21

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