SERIES - p-seriesΣ1/np, convergent if p> 1, divergent if p≤1. Geometric seriesΣarn-1or Σarn, converges if |r| < 1, diverges when |r| ≥1If 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 Σanhas some negative terms, then we can apply the Comparison Testto |Σan| and test for Absolute Convergence. If anis of the form (bn)n, then the Root Testmay be useful. If an= f(n), where ∫∞1)(dxxfis easily evaluated, then the Integral Testis effective (if hypotheses of test are satisfied)COMPARISON TESTS– Suppose that Σanand Σbn are series with positive terms.Direct comparison test (a) if Σbnis convergent and an ≤bn for all n, then Σan is also convergent. (b) if Σbn is divergent and an ≥bnfor all n, then Σan is also divergent.Limit comparison test (c) If 0lim=∞→cbannn, then either both series converge or both diverge.