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Summary of Series Tests 2 - Summary of Convergence Tests...

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Summary of Convergence Tests for Infinite Series Test Series Converges if Diverges if Comment Divergence X n =1 a n lim n →∞ a n 6 = 0 cannot be used to show convergence Geometric Series X n =0 ar n | r | < 1 | r | ≥ 1 Sum: S = a 1 - r p -Series X n =1 1 n p p > 1 p 1 Integral X n =1 a n Z 1 f ( x ) dx Z 1 f ( x ) dx f is continuous, positive, a n = f ( n ) 0 converges diverges and decreasing 0 a n b n 0 b n a n Comparison X n =1 a n and X n =1 b n and X n =1 b n a n , b n positive converges diverges a n , b n positive lim n →∞ a n b n > 0 lim n →∞ a n b n > 0 Test fails if Limit Comparison X n =1 a n and and lim n →∞ a n b n = 0 X n =1 b n converges X n =1 b n diverges or lim n →∞ a n b n = Alternating Series X n =1 ( - 1) n - 1 b n b n +1 b n lim n →∞ b n 6 = 0 b n positive lim n →∞ b n = 0 Absolute Convergence X n =1 a n X n =1 | a n | cannot be used to show divergence converges Ratio X n =1 a n lim n →∞ a n +1 a n < 1 lim n →∞ a n +1 a n > 1 Test fails if lim
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