Theorem. The Integral Test. Let

be a series such that there exists a function f with the following properties:

  1. an = f(n)
  2. f is continuous, positive and decreasing.
Then
  • If the integral

    converges, then the series

    converges.

  • If the integral

    diverges, then the series

    diverges.

Discussion [Using Flash] [Using Java]

Examples.


  1. Discussion [Using Flash] [Using Java]


  2. Discussion [Using Flash] [Using Java]


  3. Discussion [Using Flash] [Using Java]

Theorem. p-Series Test. The p-series

is convergent if p > 1 and is divergent if < 1.

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Estimating using the Integral Test.