The use comparison theorem is a mathematical tool used to determine the convergence or divergence of an improper integral by comparing it to a known integral with known convergence or divergence.
The comparison theorem states that if an integrand is always less than or equal to another integrand on a given interval, and the known integral converges, then the original integral must also converge. Similarly, if the integrand is always greater than or equal to another integrand on a given interval, and the known integral diverges, then the original integral must also diverge.
The comparison theorem provides a quick and easy way to determine the convergence or divergence of an integral without having to evaluate the integral directly. It also allows for the use of known integrals to make a determination, rather than relying solely on the properties of the integrand.
To use the comparison theorem, you must first find a known integral that is either always greater than or equal to, or always less than or equal to the original integral on the given interval. Then, you can compare the known integral's convergence or divergence to make a determination for the original integral.
To use the comparison theorem for this integral, we can compare it to the integral of 1/\sqrt{x} from 0 to 1. By noting that e^{-x} is always greater than or equal to 1 for x greater than or equal to 0, we can say that e^{-x}/\sqrt{x} is always greater than or equal to 1/\sqrt{x}, and thus the original integral must diverge since the integral of 1/\sqrt{x} from 0 to 1 is known to diverge. Therefore, we can conclude that the integral of e^{-x}/\sqrt{x} from 0 to 1 also diverges.