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 2 answersLast reply by: Jimmy TrinhSun May 3, 2015 6:55 PMPost by Angela Patrick on March 30, 2014I believe in example 2, when he says hat the series diverges it's an incorrect statement because if the test fails it should be a no conclusion situation

### Alternating Series

• Alternating negative and positive components
• Alternating series converges if:
• If given terms of a series, solve for proper series notation first ()

### Alternating Series

Does the series ∑[(( − 1)n + 1)/(n + 1)] converge?
• Check if an + 1< an
• an + 1 = [1/(n + 2)],an = [1/(n + 1)]
• [1/(n + 2)] < [1/(n + 1)]
• an + 1< an
Find limn → ∞ anlimn → ∞ [1/(n + 1)] = 0 Thus the series converges
Does the series ∑[(2( − 1)n + 1)/(n − 3)] converge?
• Check if an + 1< an
• an + 1 = [2/(n − 2)],an = [2/(n − 3)]
• [2/(n − 2)] < [2/(n − 3)]
• an + 1< an
Find limn → ∞ anlimn → ∞ [2/(n − 3)] = 0 Thus the series converges
Does the series ∑[(( − 1)n + 1)/(n2)] converge?
• Check if an + 1< an
• an = [1/(n2)],an + 1 = [1/((n + 1)2)]
• [1/((n + 1)2)] < [1/(n2)]
• an + 1< an
Find limn → ∞ anlimn → ∞ [1/(n2)] = 0 Thus the series converges
Does the series ∑[(n3( − 1)n + 1)/(n2)] converge?
• Check if an + 1< an
• an = [(n3)/(n2)]
• = n
n < [((n + 1)3)/((n + 1)2)]an + 1> an Thus the series diverges
Does the series [1/4] − [2/5] + [1/2] − [4/7] + ... + [(( − 1)n + 1n)/(n + 3)] converge?
• Determine an + 1
• an = [n/(n + 3)],an + 1 = [(n + 1)/(n + 4)]
a5 = [5/8] = 0.625a6 = [6/9] = 0.666...a6> a5, thus the series diverges
Does the series ∑[(( − 1)n + 1)/(3n)] converge?
• Determine an + 1
• an = [1/(3n)],an + 1 = [1/(3n + 1)]
• Try an arbitrary positive integer for n, such as n = 1
• a1 = [1/3] = 0.333...
• a2 = [1/9] = 0.111....
• a2< a1
Find limn → ∞ anlimn → ∞ [1/(3n)] = 0 Thus the series diverges
Does the series ∑[(( − 1)n + 1n!)/(2n)] converge?
• Determine an + 1
• an = [n!/(2n)],an + 1 = [(( n + 1 )!)/(2n + 1)]
a2 = [2/4] = 0.5a3 = [6/8] = 0.75a3> a2, thus the series diverges
Does the series ∑[(( − 1)n + 12n)/n!] converge?
• Determine an + 1
• an = [(2n)/n!],an + 1 = [(2n + 1)/(( n + 1 )!)]
• Try an arbitrary positive integer for n, such as n = 2
• a2 = [4/2]
• a3 = [8/6]
• a3< a2
Find limn → ∞ an using the Ratio Test
limn → → ∞ [(2n)/n!] = 0 Thus the series converges
Does the series ∑( − 1)n + 12( [1/3] )n − 1 converge?
r = − [1/3]r < 1 Thus the series converges
What is the sum of the ∑( − 1)n + 12( [1/3] )n − 1 ?
Apply the Sum equation
Sum = [a/(1 − r)] = [2/(1 + [1/3])] = [2/([4/3])] = [6/4]

*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.