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Strategy for using Infinite Series Test

# Strategy for using Infinite Series Test - SUMMARY OF...

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SUMMARY OF INFINITE SERIES TESTS 1. Definition of the sum of a series. Let s n = a 1 + a 2 + · · · + a n . If lim n →∞ s n exists and equals s then n =1 a n is convergent and its sum is s . 2. Well understood series. ( geometric series ) The series a + ar + ar 2 + ar 3 + · · · is convergent if | r | < 1 and the sum is a/ (1 - r ) and the series is divergent with no sum if | r | ≥ 1. ( p series ) The series n =1 1 n p is convergent if p > 1 and divergent if p 1. ( Telescoping series ) For some series n =1 a n , a partial fraction decomposition of a n results in a telescoping sum of s n making it easy to determine the sum of the series. 3. Tests for positive term series only. Suppose n =1 a n and n =1 b n are positive term series. ( limit comparison test ) If a n b n for large n , then n =1 a n and n =1 b n have the same behavior. ( comparison test ) Suppose a n b n for all n .
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