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8.4 day 2 Tests for Convergence. Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2006. Riverfront Park, Spokane, WA. If then gets bigger and bigger as , therefore the integral diverges .
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8.4 day 2 Tests for Convergence Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2006 Riverfront Park, Spokane, WA
If then gets bigger and bigger as , therefore the integral diverges. If then b has a negative exponent and , therefore the integral converges. Review: (P is a constant.)
to for positive values of x. For Does converge? Compare:
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: converges if converges. 1 diverges if diverges. 2 page 438:
The maximum value of so: on Since converges, converges. Example 7:
for positive values of x, so: on Since diverges, diverges. Example 7:
Does converge? As the “1” in the denominator becomes insignificant, so we compare to . Since converges, converges. If functions grow at the same rate, then either they both converge or both diverge.
