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SeriesSummary

# SeriesSummary - Form Converges if Diverges if Sum of Series...

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Summary of Tests for Series http://www-scf.usc.edu/ ~ bouma http://college.usc.edu/math126 Name Conditions a n convergent if. .. a n divergent if. .. Inconclusive if. .. Sum of Series? Test for Di- vergence n/a n/a lim n →∞ a n 6 = 0 lim n →∞ a n = 0 n/a Integral Test a n = f ( x ) S n + R n +1 S S n + R n 1. f ( x ) positive Z 1 f ( x ) d x converges Z 1 f ( x ) d x diverges n/a Z n +1 f ( x ) R n Z n f ( x ) 2. f ( x ) continuous 3. f ( x ) decreasing Comparison Test small a n big X n =1 big converges X n =1 small diverges small converges or big diverges n/a Limit 1. a n 0 0 < C < and b n conv. 0 < C < and b n div. C = 0, b n div. Comparison 2. b n 0 or or or n/a Test 3. lim n →∞ a n b n = C C = 0 and b n conv. C = and b n div. C = , b n conv. Alternating b n 0 Series Test b n +1 b n (decr.) takes the form X n =0 ( - 1) n b n n/a n/a | R n | = | S - S n | ≤ b n +1 lim n →∞ b n = 0 Ratio Test lim n →∞ ± ± ± ± a n +1 a n ± ± ± ± = L L < 1, absolutely L > 1 L = 1 n/a Root Test lim n →∞ n q ± ± a n ± ± = L L < 1, absolutely L > 1 L = 1 n/a Special Series Name
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Unformatted text preview: Form Converges if. .. Diverges if. .. Sum of Series? Geometric Series ∞ X n =1 a r n-1 = ∞ X n =0 a r n | r | < 1 | r | ≥ 1 a 1-r p-Series ∞ X n =1 1 n p p > 1 p ≤ 1 n/a Definitions Absolute convergence. A series ∑ a n is called absolutely convergent if the series ∑ | a n | converges by one of the above tests. Conditional convergence. A series ∑ a n is called conditionally convergent if the series ∑ a n converges by one of the above tests, but the series ∑ | a n | diverges. Note that if a series is absolutely convergent, then it is also conditionally convergent, but not the other way around....
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