# Strategy for Testing Series

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• p-series
• convergent if p>1
• divergent if p≤1
1. or
• geometric series
• converges if |r|<1
• diverges if |r|≥1
2. An is a rational function or algebraic function of n (involving roots of polynomials)
• comparison test with p-series
• the 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.
3. If series is similar to geo series
• comparison test with geo series
• NOTE: Comparison test only applies to series with positive terms. If it has negative terms, test for absol convergence.
4. Test for Divergence
5. or
• Alternating Series Test
• (i) bn+1 ≤  bn for all n, and
• (ii) ,then the series is convergent.
6. 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.
7. Harmonic series. Diverges
8. 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.
9. If , where is easily evalutated
• Integral Test
• If if f is cts, positive and decreasing fcn on [1,∞), and an=f(n), then the series is convergent IFF is convergent
10. If series is similar to p-series or geometric series
• Limit Comparison Test
• Given the series Σan, Σbn with positive terms,if , where c > 0 and finite, then the two series either both converge or both diverge.
 Author: Jamie_Bee ID: 291067 Card Set: Strategy for Testing Series Updated: 2014-12-09 00:11:21 Tags: Convergence Divergence IntermediateCalculus Folders: Description: Strategy for Testing Series Show Answers: