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Calculus II - 24

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Stewart Calculus Section 11.4

Calculus II - 24

  1. 1. 11.4 The Comparison Tests The convergence of some series can be determined easily by the integral test, for example: = Some series look similar, but the integral test cannot be easily applied, for example: = +
  2. 2. The Comparison Test: ∞ ∞Suppose = and = are series withpositive terms, ∞ If = is convergent and ≤ ∞ then = is also convergent. ∞ If = is divergent and ≥ ∞ then = is also divergent. If = > is finite then they either both converge or both diverge.
  3. 3. Common series used for comparison: p-series: converges for > = diverges for = geometric series: converges for | | < = diverges for | | =
  4. 4. Ex: Test the series for convergence. = +
  5. 5. Ex: Test the series for convergence. = + < + is convergent. = So is convergent. = +
  6. 6. Ex: Test the series for convergence. =
  7. 7. Ex: Test the series for convergence. = > is divergent. = So is divergent. =
  8. 8. Ex: Test the series for convergence. =
  9. 9. Ex: Test the series for convergence. = = is convergent. = So is convergent. =
  10. 10. +Ex: Test the series for convergence. = +
  11. 11. +Ex: Test the series for convergence. = + + = + is divergent (p-test). = + So is divergent. = +
  12. 12. +Ex: Test the series for convergence. = +
  13. 13. +Ex: Test the series for convergence. = + + = + is convergent (geometric). = + So is convergent. = +

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