Q. 1TB
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?
Step-by-Step Solution
VerifiedIf \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.
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.
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.
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.