Unformatted text preview:  < 1; a pseries converges if p > 1.) The limit comparison test is more “versatile” than the basic comparison test. With either type of comparison test, you have to choose b n . In the limit comparison test, L = lim a n b n . If L 6 = 0 and L 6 = ∞ , the two series ∑ a n and ∑ b n behave the same. (It’s irrelevant whether L < 1 or L > 1.) The integral test is useful for series like ∑ 1 n ln n or ∑ 1 n (ln n ) 2 because the substitution u = ln x will help in the corresponding integral. The ratio test is useful for series containing factorials, and for power series. In the ratio test, L = lim  a n +1 a n  . If L < 1, the series converges absolutely (this includes L = 0). If L > 1, the series diverges (this includes L = ∞ ). The only value of L that gives you no information is L = 1. 1...
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 Spring '08
 wang
 Calculus, Mathematical Series

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