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Review of Series and Sequences - MATH 191[1]

Review of Series and Sequences - MATH 191[1] - SERIES...

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SERIES - p -series Σ 1/ n p , convergent if p > 1, divergent if p 1. Geometric series Σ ar n -1 or Σ ar n , converges if | r | < 1, diverges when | r | 1 If the series has a form that is similar to a p -series or a geometric series, then one of the comparison tests should be considered. If Σ a n has some negative terms, then we can apply the Comparison Test to | Σ a n | and test for Absolute Convergence . If a n is of the form ( b n ) n , then the Root Test may be useful. If a n = f ( n ), where 1 ) ( dx x f is easily evaluated, then the Integral Test is effective (if hypotheses of test are satisfied) COMPARISON TESTS – Suppose that Σ a n and Σ b n are series with positive terms. Direct comparison test (a) if Σ b n is convergent and a n b n for all n, then Σ a n is also convergent. (b) if Σ b n is divergent and a n b n for all n, then Σ a n is also divergent. Limit comparison test (c) If 0 lim = c b a n n n , then either both series converge or both diverge.
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