Question

Let f(x) be a function which, for x>0, is continuous and satisfies 1/x^2 ≤ f(x) ≤ 1/(x^2). For each of the following improper integrals, indicate whether the integral converges (in which case enter CONVERGES into the answer blank), diverges (in which case enter DIVERGES into the answer blank), or whether there is Not Enough Information to tell (in which case enter NEI into the answer blank). Hint: do comparisons using what you know about the convergence and divergence of ∫∞(1/x^n)dx for various values of n. (a) Does ∫∞(1/f(x)√x)dx converge, does it diverge, or is there not enough information to tell? (b) Does ∫∞(0.000001/f(x))dx converge, does it diverge, or is there not enough information to tell?

          Let f(x) be a function which, for x>0, is continuous and satisfies 1/x^2 ≤ f(x) ≤ 1/(x^2). For each of the following improper integrals, indicate whether the integral converges (in which case enter CONVERGES into the answer blank), diverges (in which case enter DIVERGES into the answer blank), or whether there is Not Enough Information to tell (in which case enter NEI into the answer blank). Hint: do comparisons using what you know about the convergence and divergence of ∫∞(1/x^n)dx for various values of n.

(a) Does ∫∞(1/f(x)√x)dx converge, does it diverge, or is there not enough information to tell?
(b) Does ∫∞(0.000001/f(x))dx converge, does it diverge, or is there not enough information to tell?

Added by Courtney P.

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