1. Improper Integrals Part 2 Tests for Convergence

2. Review: If then gets bigger and bigger as, therefore the integral diverges. If then b has a negative exponent and, therefore the integral converges. (P is a constant.)

3. Converges

4. Does converge? Compare: to for positive values of x. For

5. Since is always below, we say that it is “bounded above” by. Since converges to a finite number, must also converge!

6. Direct Comparison Test: Let f and g be continuous on with for all, then: 2 diverges if diverges. 1 converges if converges. page 438:

7. Example 7: The maximum value of so: on Since converges, converges.

8. Example 7: for positive values of x, so: Since diverges, diverges. on

9. If functions grow at the same rate, then either they both converge or both diverge. Does converge? As the “1” in the denominator becomes insignificant, so we compare to. Since converges, converges.

10. Of course

11. 