# How do you test the improper integral #int x^(-1/2) dx# from #[1,oo)# and evaluate if possible?

Based on the integral test the convergence of the integral:

is equivalent to the convergence of the series:

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To test the improper integral ( \int_1^\infty x^{-1/2} , dx ), you can check for convergence or divergence using the limit comparison test.

Given the integral ( \int_1^\infty x^{-1/2} , dx ), set up a comparison integral with a function ( g(x) ) that is easier to integrate and whose convergence or divergence is known.

One common choice is to compare with ( g(x) = x^{-1/2} ), which is also the integrand function.

Apply the limit comparison test by computing the following limit:

[ \lim_{x \to \infty} \frac{f(x)}{g(x)} ]

If the limit is finite and positive, then both integrals either converge or diverge. If the limit is zero or infinite, then the integral ( \int_1^\infty x^{-1/2} , dx ) behaves the same as the comparison function.

In this case, since ( f(x) = x^{-1/2} ) and ( g(x) = x^{-1/2} ), the limit simplifies to 1, which is finite and positive.

Therefore, the integral ( \int_1^\infty x^{-1/2} , dx ) converges.

To evaluate the integral, you can integrate ( x^{-1/2} ) from 1 to ( \infty ):

[ \int_1^\infty x^{-1/2} , dx = \lim_{b \to \infty} \int_1^b x^{-1/2} , dx ]

[ = \lim_{b \to \infty} \left[2x^{1/2}\right]_1^b ]

[ = \lim_{b \to \infty} \left(2\sqrt{b} - 2\right) ]

Since ( \lim_{b \to \infty} \sqrt{b} = \infty ), the limit becomes ( \infty - 2 ).

So, the value of the integral is ( \infty ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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