Unformatted text preview: f ( x ) ≥ g ( x ) ≥ for x ≥ a . • If R ∞ a f ( x ) dx is convergent then R ∞ a g ( x ) dx is convergent. • If R ∞ a g ( x ) dx is divergent then R ∞ a f ( x ) dx is divergent. Use the Comparison Theorem to determine whether the integral is convergent or divergent. You do not have to evaluate the integrals. You might have a look at Example 9 and Example 10 in Section 7.8 in the course textbook. (a) R ∞ 1 2+ e-x x dx Hint: Try to ﬁnd some relation between 2+ e-x x and 1 x for x ≥ 1. (b) R ∞ ( x 3 e-2 x ) dx Hint: Show that x 3 e-2 x ≤ e-x for suﬃciently large x . You might use the fact that x ≥ 3 ln x for all x ≥ 5....
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- Fall '08
- dx, Riemann integral, Comparison Theorem, Koray Karabina, [email protected]