Convergence Tests for Infinite Series - CONVERGENCE TESTS FOR INFINITE SERIES The geometric series in that case its sum is n n=0 ar = a a n n=0 ar = 1-r

# Convergence Tests for Infinite Series - CONVERGENCE TESTS...

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CONVERGENCE TESTS FOR INFINITE SERIES The geometric series n =0 ar n = a + ar + ar 2 + ar 3 + · · · is convergent if | r | < 1 and in that case its sum is n =0 ar n = a 1 - r . The geometric series is divergent if | r | ≥ 1. Test for Divergence If lim n →∞ a n does not exist or if lim n →∞ a n = 0 then the series n =1 a n is divergent . If lim n →∞ a n = 0, then the test is inconclusive. The Integral Test Suppose f ( x ) is a positive, continuous and decreasing function on an interval [ a, ) such that f ( n ) = a n . Then (i) if a f ( x ) dx is convergent, then n =1 a n is convergent, and (ii) if a f ( x ) dx is divergent, then n =1 a n is divergent. The p-series n =1 1 n p is convergent if p > 1 and divergent if p 1. The Comparison Test Suppose that n =1 a n and n =1 b n are series with positive terms and a n b n eventually. Then (i) if n =1 b n is convergent, then n =1 a n is convergent also, and (ii) if n =1 a n is divergent, then n =1 b n is divergent also. The Limit Comparison Test Suppose that n =1 a n and n =1 b n are series with positive terms. If lim n →∞ a n b n = c where c is a finite positive number, then either both series converge or both diverge.