Strategy for Testing Series

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Author:
Jamie_Bee
ID:
291067
Filename:
Strategy for Testing Series
Updated:
2014-12-08 19:11:21
Tags:
Convergence Divergence IntermediateCalculus
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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.

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