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Understanding Convergence and Divergence in Integrals: A Mathematical Framework

This resource explores the concepts of convergence and divergence in integral calculus, illustrating how to determine the behavior of integrals based on their expressions. We discuss key tests, such as the Direct Comparison Test, and present examples demonstrating positive values of x, bounded functions, and the implications of negative exponents. By analyzing ratios and behavior as they grow, we provide a comprehensive overview that helps students assess integral convergence and divergence effectively. Ideal for high school students studying calculus.

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Understanding Convergence and Divergence in Integrals: A Mathematical Framework

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  1. 8.4 day 2 Tests for Convergence Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2006 Riverfront Park, Spokane, WA

  2. 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.)

  3. Converges

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

  5. For 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: converges if converges. 1 diverges if diverges. 2 page 438:

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

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

  9. 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.

  10. Of course

  11. Of course p

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