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Unformatted text preview: f ne Z ∞ −∞ f ( x ) dx = Z a −∞ f ( x ) dx + Z ∞ a f ( x ) dx where a is any real number. Comparison Test for Improper Integrals Suppose that f and g are continuous functions with f ( x ) ≥ g ( x ) ≥ for x ≥ a . ( a ) If Z ∞ a f ( x ) dx is convergent then Z ∞ a g ( x ) dx is convergent. ( b ) If Z ∞ a g ( x ) dx is divergent then Z ∞ a f ( x ) dx is divergent . Useful Result for Using Comparison Test Z ∞ 1 1 x p dx converges if and only if p > 1 . Useful Inequalities: 1 1 + y p ≤ 1 y p , y ≥ 1 1 1 + y p ≥ 1 y p + y p = 1 2 y p , y ≥ 1 2...
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This note was uploaded on 04/27/2010 for the course STAT 330 taught by Professor Paulasmith during the Fall '08 term at Waterloo.
- Fall '08