Contents
What is the comparison theorem for integrals?
The comparison theorem for improper integrals is very similar to the comparison test for convergence that you’ll study as part of Sequences & Series. It allows you to draw a conclusion about the convergence or divergence of an improper integral, without actually evaluating the integral itself.
How do you choose comparison functions?
When choosing a function for direct comparison, you want it to have certain qualities:
- Its integral should be known to converge on some interval [a,∞).
- It should be greater than your function of interest, if it converges, or less than your function of interest, if it diverges.
How do you know if an integral converges?
If the integration of the improper integral exists, then we say that it converges. But if the limit of integration fails to exist, then the improper integral is said to diverge.
What are the two types of improper integrals?
There are two types of improper integrals:
- The limit a or b (or both the limits) are infinite;
- The function f(x) has one or more points of discontinuity in the interval [a,b].
How do you tell if a function converges or diverges?
convergeIf a series has a limit, and the limit exists, the series converges. divergentIf a series does not have a limit, or the limit is infinity, then the series is divergent. divergesIf a series does not have a limit, or the limit is infinity, then the series diverges.
How do you do a limit comparison test?
The Limit Comparison Test
- If the limit of a[n]/b[n] is positive, then the sum of a[n] converges if and only if the sum of b[n] converges.
- If the limit of a[n]/b[n] is zero, and the sum of b[n] converges, then the sum of a[n] also converges.
How do you prove a limit comparison test?
If c is positive (i.e. c>0 ) and is finite (i.e. c<∞ ) then either both series converge or both series diverge. The proof of this test is at the end of this section.
What does the limit comparison test tell us?
The limit comparison test shows that the original series is convergent. The limit comparison test does not apply because the limit in question does not exist. The comparison test can be used to show that the original series converges. The comparison test can be used to show that the original series diverges.
How do you know if a function converges or diverges?
How do you tell if an integral is proper or improper?
Integrals are improper when either the lower limit of integration is infinite, the upper limit of integration is infinite, or both the upper and lower limits of integration are infinite.