Final Answer:
The improper integral ∫₁^∞ dx/√(x) converges.
It converges to 2.
Step-by-step explanation:
To evaluate the convergence of the improper integral, we can analyze its behavior as x approaches infinity. The integral ∫₁^∞ dx/√(x) can be rewritten as ∫₁^∞ x^(-1/2) dx.
The integral converges because the exponent -1/2 in the integrand is less than 1. To find what it converges to, we can integrate the expression. The antiderivative of x^(-1/2) is 2x^(1/2), and evaluating this from 1 to ∞ gives the result of 2.
Therefore, the given improper integral converges, and its value is 2. This means that as x approaches infinity, the area under the curve 1/√(x) from 1 to ∞ is finite and equals 2. The convergence is a result of the function approaching zero as x goes to infinity, making the integral finite.