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Unformatted text preview: Series Summary A sequence is an ordered list of numbers: { a n } = { a 1 ,a 2 ,a 3 ,... } , and a series is the sum of those numbers: ∞ X n =1 a n = a 1 + a 2 + a 3 + ··· . In either case, we want to determine if the sequence converges to a finite number or diverges and if the series converges to a finite number or diverges. If the series converges, that means that a sum of infinitely many numbers is equal to a finite number! If the sequence { a n } diverges or converges to anything other than 0, then the series ∑ a n diverges. If the sequence { a n } converges to 0, then the series ∑ a n may converge or may diverge. For any given series ∑ a n there are two associated sequences: the sequence of terms { a n } and the sequence of partial sums { s n } , where s n = a 1 + a 2 + ... + a n . If ∑ a n = L , then lim n →∞ a n = 0 (as stated above) and lim n →∞ s n = L . 1 When can we calculate the sum of a series? Unfortunately, we are unable to compute the exact sum of a series in most cases. However, there are a few examples that can be computed....
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This note was uploaded on 02/17/2012 for the course MTH 133 taught by Professor Staff during the Fall '08 term at Michigan State University.
 Fall '08
 STAFF
 Calculus, Infinite Series

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