Introduction to :

We often use integrals of the functions , for various values of p, to help determine whether other integrals converge or diverge.

You already did the work to show this, so we'll just summarize the results. Assuming p is greater than 0 (because otherwise the exponents do weird things),

  •  converges if p < 1 and diverges otherwise.
  •  converges if p > 1 and diverges otherwise.

This is often called the p-test for improper integrals.

Practice:

Example 1

Use the <em>p</em>-test to determine if

converges or diverges.


Exercise 1

Use the p-test to determine if the integral converges or diverges.


Exercise 2

Use the p-test to determine if the integral converges or diverges.


Exercise 3

Use the p-test to determine if the integral converges or diverges.


Exercise 4

Use the p-test to determine if the integral converges or diverges.


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