checking_series_convergence

checking_series_convergence - f ( x ); be sure to check...

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Procedure for Determining Convergence Divergence Test If lim n →∞ a n 6 = 0, then X n =1 a n diverges. Geometric Series X n =1 ar n - 1 = a + ar + ar 2 + ... = a 1 - r if | r | < 1 diverges if | r | ≥ 1 p-Series X n =1 1 n p ( converges if p > 1 diverges if p 1 Analyze X n =1 | a n | This is a positive series (all terms 0), which means that we can usually all of the tests listed below (be careful with the ratio test if some terms are zero ). When X n =1 | a n | converges, we say that X n =1 a n converges absolutely (absolute convergence implies convergence, but not vice versa). Comparison Tests p -series or geometric series. Ratio Test Use when terms are built out of n ! and/or c n for constant c ; fails when applied to p -series look-alikes. Root Test Use when a n = ( b n ) n . Integral Test Use if a n = f ( n ) for n = 1 , 2 ,... and you can integrate
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Unformatted text preview: f ( x ); be sure to check these 3 conditions first: f ( x ) must be positive, continuous & decreasing. Alternating Series Test If ∞ X n =1 | a n | = ∞ X n =1 b n diverges, it is still possible for ∞ X n =1 (-1) n b n or ∞ X n =1 (-1) n-1 b n to converge conditionally . Remember to check both conditions necessary for convergence: lim n →∞ b n = 0 and b n +1 ≤ b n (decreasing). If All Else Fails Might need to look at partial sums S n = a 1 + ... + a n . For example, if the series is telescoping, computing ∞ X n =1 a n = lim n →∞ ( a 1 + ... + a n ) = lim n →∞ S n may be the only way to determine convergence (or divergence)....
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This note was uploaded on 09/21/2009 for the course MATH 1234 taught by Professor Egcdd during the Spring '09 term at Aarhus Universitet.

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