spaniks said:Homework Statement
use the comparison theorem to show that the integral of e^(-x^2) from 0 to infinity is convergent.
Homework Equations
None
The Attempt at a Solution
In class we have never dealt with using the comparison theorem with the exponential function so I was not sure what I function I would compare it to in order to solve this problem. Could I compare it to something like e^(-x)?
The comparison theorem for integrals states that if two functions have the same behavior near a given point, and one is integrable while the other is not, then the integral of the first function must also be integrable.
Yes, the comparison theorem can be used to prove both convergence and divergence of an integral. If the integrals of two functions can be compared, and one is known to converge while the other diverges, then the same must be true for the original integral.
The comparison theorem is a powerful tool in determining the convergence or divergence of an integral. It allows us to compare a given integral to a known integral and make conclusions about its behavior without having to evaluate the integral directly.
Yes, there are some limitations to using the comparison theorem. It can only be used when the functions being compared have the same behavior near a given point. It also cannot be used if the functions have different behaviors at the endpoints of the interval of integration.
Yes, the comparison theorem can be used for improper integrals. As long as the functions being compared have the same behavior near the point of integration, the comparison theorem can be applied to determine the convergence or divergence of the improper integral.