# Strategy for Testing Series

 p-seriesconvergent if p>1 divergent if p≤1 or geometric seriesconverges if |r|<1diverges if |r|≥1 An is a rational function or algebraic function of n (involving roots of polynomials) comparison test with p-seriesthe value of p should be chosen by keeping only the highest powers of n in the numerator and denomiator(i) Convergence: If Σbn is convergent, and an ≤ bn for all n, then Σan is convergent;(ii) Divergence: If Σbn is divergent, and an ≥ bn for all n, then Σan is divergent.NOTE: Comparison test only applies to series with positive terms. If it has negative terms, test for absol convergence. If series is similar to geo series comparison test with geo seriesNOTE: Comparison test only applies to series with positive terms. If it has negative terms, test for absol convergence. Test for Divergence or Alternating Series Test(i) bn+1 ≤  bn for all n, and(ii) ,then the series is convergent. series with factorials, products (a constant raised to the nth power Ratio test(i) If  < 1, then the series Σan is absolutely convergent(and therefore convergent)(ii) If >1or , then Σan is divergent(iii) If =1, the Ratio Test is inconclusive. NOTE: as n→ ∞ for all p-series. Harmonic series. Diverges an is of the form (bn)n Root Test(i) If < 1, then Σan is absolutely convergent (and therefore convergent)(ii) If > 1, or , then Σan is divergent(iii) If , the Root Test is inconclusive. If , where is easily evalutated Integral TestIf if f is cts, positive and decreasing fcn on [1,∞), and an=f(n), then the series is convergent IFF is convergent If series is similar to p-series or geometric series Limit Comparison TestGiven the series Σan, Σbn with positive terms,if , where c > 0 and finite, then the two series either both converge or both diverge. AuthorJamie_Bee ID291067 Card SetStrategy for Testing Series DescriptionStrategy for Testing Series Updated2014-12-09T00:11:21Z Show Answers