Unformatted text preview: a N +1 + ∑ ∞ m =2 a N + m . Case 2 : l > 1. This time choose ² > 0 small enough so that q := l² > 1. Then there is some N such that if n > N ,  a n +1   a n  > q In particular,  a N +2  > q  a N +1  ,  a N +3  > q  a N +2  > q 2  a N +1  , . . . ,  a N + m  > q m1  a N +1  , ∀ m > 1 . Since q > 1, it is clear that it is impossible that lim n →∞ a n = 0 (in fact, lim n →∞  a n  = + ∞ !), which is necessary for convergence of the series ∑ ∞ n =1 a n (see Theorem 6 . 9). 1...
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 Summer '06
 DeLeenheer
 Mathematical Series, Patrick De Leenheer, an, comparison test yields

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