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TestsForSeries - A Alaca MATH 1005A MATH 1005 A WINTER 2007...

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A. Alaca MATH 1005A Winter 2007 1 MATH 1005 A WINTER 2007 LECTURE SLIDES Prepared by Ay¸ se Alaca Last modified: February 5, 2007
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A. Alaca MATH 1005A Winter 2007 2 INFINITE SEQUENCES AND SERIES Definition: Let n =1 a n be an infinite series and let s n = n i =1 a i . If the sequence ( s n ) is convergent and lim n -→∞ s n = s , then we say that n =1 a n is convergent, and we call s the sum of the series, and we write n =1 a n = s. lim n -→∞ s n = s ⇐⇒ n =1 a n = s. If ( s n ) is divergent, then n =1 a n is also divergent. Definition: The series given by n =1 ar n - 1 = a + ar + ar 2 + · · · + ar n - 1 + · · · , a = 0 . is called the geometric series with ratio r . n =1 ar n - 1 = lim n -→∞ s n = lim n -→∞ a (1 - r n ) 1 - r = a 1 - r , if - 1 < r < 1 divergent , if r > 1 or r ≤ - 1 . = a 1 - r , if | r | < 1 divergent , if | r | ≥ 1 . Theorem: If the series n =1 a n is convergent, then lim n -→∞ a n = 0. Theorem ( n th term for divergence): If lim n -→∞ a n does not exist or lim n -→∞ a n = 0 = n =1 a n is divergent.
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