(Leads into Section 8.3 for Series!!!)

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Presentation transcript:

(Leads into Section 8.3 for Series!!!) 7.8 Comparison Test for Improper Integrals (Leads into Section 8.3 for Series!!!) Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2006 Riverfront Park, Spokane, WA

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

Converges

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

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

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

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

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

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

Of course p