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Positive Series:

Section 10.3 – The Integral Test. Positive Series:. A series whose elements are non-negative . The Integral Test:. Let f be a continuous, positive and decreasing function and for all positive integers, then converges iff the improper integral converges.

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Positive Series:

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  1. Section 10.3 – The Integral Test Positive Series: A series whose elements are non-negative. The Integral Test: Let f be a continuous, positive and decreasing function and for all positive integers, then converges iff the improper integral converges If the improper integral diverges, then the series diverges.

  2. Section 10.3 – The Integral Test Example:

  3. Section 10.3 – The Integral Test Positive Series p-Series p is a constant If p is  1, then the series converges. If p is  1, then the series diverges.

  4. Section 10.3 – The Integral Test Positive Series Examples: converges diverges

  5. Section 10.3 – The Integral Test Positive Series Harmonic Series diverges

  6. Section 10.3 – The Integral Test

  7. Section 10.4 – Comparison Tests Positive Series Comparison Test (Ordinary Comparison Test) Let be a positive series. If is a positive convergent series and ,then the series converges. is a positive divergent series and ,then If the series diverges.

  8. Section 10.4 – Comparison Tests Positive Series Examples: is a geometric series. is a convergent geometric series. is a convergent series.

  9. Section 10.4 – Comparison Tests Positive Series Limit Comparison Test Let be a positive series and is a rational expression. Choose to be a positive series. If , then will converge or diverge depending on provided . will converge if converges provided . will diverge if diverges provided .

  10. Section 10.4 – Comparison Tests Positive Series Examples: is a p-series. is div. is div. is div.

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