Chapter 6: Improper Integrals

Question 1

For what values of p is

∫₁^∞ 1/(x^p) dx

convergent and divergent?

Answer 1

Convergent if p > 1.

Divergent if p ≤ 1.

Question 2

Suppose that f and t are continuous functions with f(x) ≥ g(x) ≥ 0 for x ≥ a.

If ∫_a^∞ g(x) dx is divergent,

Answer 2

then ∫_a^∞ f(x) dx is divergent.

Question 3

Suppose that f and t are continuous functions with f(x) ≥ g(x) ≥ 0 for x ≥ a.

If ∫_a^∞ f(x) dx is convergent,

Answer 3

then ∫_a^∞ g(x) dx is convergent.