iv In the case a n cos n\u03c0 parenleftbigg 5 n 7 n 4 parenrightbigg n 1 n

Iv in the case a n cos nπ parenleftbigg 5 n 7 n 4

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Unformatted text preview: (iv) In the case a n = cos( nπ ) parenleftbigg 5 n 7 n + 4 parenrightbigg n = ( − 1) n parenleftbigg 5 n 7 n + 4 parenrightbigg n we apply the root test to the series ∑ n | a n | . Then lim n →∞ | a n | 1 /n = lim n →∞ 5 n 7 n + 4 = 5 7 < 1 . Thus both of the series ∞ summationdisplay n = 1 a n , ∞ summationdisplay n = 1 | a n | converge. (v) The series ∞ summationdisplay n = 1 a n , a n = ( − 2) n is a geometric series with r = − 2 < − 1 ; on the other hand, the series ∞ summationdisplay n = 1 | a n | , | a n | = (2) n is a geometric series with r = 2 > 1 . Thus neither of ∞ summationdisplay n =1 a n , ∞ summationdisplay n =1 | a n | converges. Consequently, of the five given infinite se- ries, only ∞ summationdisplay k =1 ( − 1) k 4 7 k log( k ) + 8 converges conditionally but not absolutely. keywords:...
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  • Fall '11
  • Gramlich
  • Accounting, Mathematical Series, lim, n=1

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