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### 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/(n2)] converge absolutely?
• Determine if ∑n = 1 | an | converge
• n = 1 | ( − 1)n + 1[1/(n2)] | = ∑n = 1 | [1/(n2)] |
Apply P - series properties
n = 1 | [1/(n2)] | = 0 Thus the series converges completely
Does the series ∑n = 1 ( − 1)n + 1[(n − 1)/(n2 + 2)] converge absolutely?
• Determine if ∑n = 1 | an | converge
• n = 1 | ( − 1)n + 1[(n − 1)/(n2 + 2)] | = ∑n = 1 | [(n − 1)/(n2 + 2)] |
• Apply Harmonic series properties
= [1/n] = diverges
Thus the series does not absolutely converge
Does the series ∑n = 1 [(( − 1)n + 1)/(√n + 3)] absolutely converge?
• Determine if ∑n = 1 | an | 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)/(3n − 1)] absolutely converge?
• Determine if ∑n = 1 | an | converge
• n = 1 | [(( − 1)n)/(3n − 1)] | = ∑n = 1 | [1/(3n − 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 | an | converge
• n = 1 | [(( − 1)n + 1)/(4√{n + 2})] | = ∑n = 1 | [1/(4√{n + 2})] |
n = 1 | [1/(4√{n + 2})] | = ∑n = 1 | [1/(( n + 2 )1/4)] | p = [1/4] < 1 = diverges
Thus the series does not absolutely converge
Does the series ∑n = 1 [(( − 1)n + 1n2)/(3√{n})] absolutely converge?
• Determine if ∑n = 1 | an | converge
= ∑n = 1 | n5/3 | = ∞ Thus the series does not absolutely converge
Does the series ∑n = 1 [(2( − 1)n(n + 2)2)/(n2 + 1)] absolutely converge?
• Determine if ∑n = 1 | an | converge
• n = 1 | [(2( − 1)n(n + 2)2)/(n2 + 1)] | = ∑n = 1 | [(2(n + 2)2)/(n2 + 1)] |
• = ∑n = 1 | [(2(n2 + 2n + 4))/(n2 + 1)] |
• = ∑n = 1 | [(2n2 + 4n + 8)/(n2 + 1)] |
• = [(2n2)/(n2)]
• = 2
Thus the series does absolutely converges
Does the series ∑n = 1 [(( − 1)n)/(3n + 1)] absolutely converge?
• Determine if ∑n = 1 | an | converge
• n = 1 | [(( − 1)n)/(3n + 1)] | = ∑n = 1 | [1/(3n + 1)] |
Let bn = [1/n]limn∞ [([1/(3n + 1)])/([1/n])] = [1/3] Thus the series absolutely converges
Does the series ∑n = 1 [(( − 1)n + 2(n − 1))/(n3)] absolutely converge?
• Determine if ∑n = 1 | an | converge
• n = 1 | [(( − 1)n + 2(n − 1))/(n3)] | = ∑n = 1 | [(n − 1)/(n3)] |
Let bn = [1/(n2)]limn∞ [([(n − 1)/(n3)])/([1/(n2)])] = 1 Thus the series absolutely converges
Does the series ∑n = 1 [(( − 1)n + 12n)/n!] absolutely converge?
• Determine if ∑n = 1 | an | converge
• n = 1 | [(( − 1)n + 12n)/n!] | = ∑n = 1 | [(2n)/n!] |
Apply the Ratio Test
n = 1 | [(2n)/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.