11.4.misc5-eg - where we have used l’Hopital’s multiple...

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Example Use the Limit Comparison Test to determine if the series converges or diverges. X n =1 2 3 n (5 n + 1) 2 . We first note that 2 3 n (5 n + 1) 2 behaves globally like (2 3 ) n (5 n ) 2 = 8 n 25 n = ± 8 25 ² n . Applying the Limit Comparison Test with ± 8 25 ² n , lim n →∞ 2 3 n (5 n +1) 2 ³ 8 25 ´ n = lim n →∞ 2 3 n (5 n + 1) 2 · 5 2 n 2 3 n = lim n →∞ 5 2 n (5 n + 1) 2 = 1
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Unformatted text preview: , where we have used l’Hopital’s multiple times. Since the limit above is finite and the series ∞ X n =1 ± 8 25 ² n is a convergent geometric series, we conclude that the original series converges as well....
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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