Question

The comparison test: In Section 5.6 we discussed the comparison test for improper integrals.

• What is the comparison test for improper integrals?
• How is the comparison test for improper integrals used to analyze the convergence or divergence of an improper integral?
• Which improper integrals are used in the comparison test?

Short answer

If \int_a^{\infty} f(x) d x converges, then \int_a^{\infty} g(x) d x converges.

If \int_a^{\infty} g(x) d x diverges, then \int_a^{\infty} f(x) d x diverges.

The limit comparison test and comparison test are used in the comparison test.

Step-by-step solution

Step 1. Comparison Test for Improper Integrals 

Comparison Test for Improper Integrals Let f(x) and g(x) be continuous functions.

Assume that f(x) \geq g(x) \geq 0 for x \geq a.

If \int_a^{\infty} f(x) d x converges, then \int_a^{\infty} g(x) d x converges.

If \int_a^{\infty} g(x) d x diverges, then \int_a^{\infty} f(x) d x diverges.

Step 2. analyze the convergence or divergence of an improper integral 

The improper integral converges if the limit exists and is a finite number. The improper integral diverges if the limit is \pm\infty or does not exist.

Suppose f(x) is continuous on the interval [a, b]

If \int_a^{\infty} f(x) d x converges, then \int_b^{\infty} f(x) d x also converges.

If \int_a^{\infty} f(x) d x diverges, then \int_b^{\infty} f(x) d x also diverges.

Step 3. Limit comparison test and Comparison test

Limit comparison test: Let b be a real number or b=\infty, let a<b. Let f and g be functions that are continuous on [a, b), let f \geq 0 there. Assume that the limit \lim _{x \rightarrow b^{-}}\left(\frac{f(x)}{g(x)}\right) exists finite, but is not equal to zero. Then the integral \int_a^b f(x) d x converges if and only if the integral \int_a^b g(x) d x converges.

Comparison test: Let b be a real number or b=\infty, let a<b. Let f and g be functions that are continuous on [a, b) and let |f| \leq g on [a, b). If \int_a^b g(x) d x converges, then also \int_a^b f(x) d x converges.