Improper Integrals Part 2 Tests for Convergence

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.)

Converges

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

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

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

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

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

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.

Of course

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