ComparisonTestImproperIntegrals - Penn State Altoona Math 141 The Comparison Tests for Improper Integrals Improper integrals of the form a f(x dx

ComparisonTestImproperIntegrals - Penn State Altoona Math...

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Penn State Altoona Math 141 The Comparison Tests for Improper IntegralsImproper integrals of the formaf(x)dxTheorem (Comparison Test).Suppose thatfandgare continuous functions on[a,), and supposethatg(x)>0for all sufficiently large values ofx.a) If|f(x)| ≤g(x)for all sufficiently large values ofx, andag(x)dx <,thenaf(x)dxconverges.b) Iff(x)g(x)for all sufficiently large values ofx, andag(x)dx=,thenaf(x)dxdiverges.Theorem (Limit Comparison Test).Suppose thatfandgare continuous functions on[a,), andsuppose thatg(x)>0for all sufficiently large values ofx.

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