How do you use the integral test to determine whether #int x^x# converges or diverges from #[1,oo)#?
The integral:
is convergent.
Based on the integral test, the convergence of the integral:
is equivalent to the convergence of the series:
the series has positive terms and we can apply the root test:
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To use the integral test to determine the convergence or divergence of the improper integral (\int_{1}^{\infty} x^{x} , dx), follow these steps:

Check for Positivity and Continuity: Ensure that the function (f(x) = x^{x}) is continuous and positive for (x \geq 1).

Integrate the Function: Attempt to integrate (f(x)) over the given interval. In this case, integrating (x^{x}) can be quite challenging.

Use Comparison Test: If integration is not feasible, try to find another function (g(x)) that is easier to integrate and which satisfies (0 \leq f(x) \leq g(x)) for (x \geq 1). Then, apply the comparison test to determine convergence or divergence of (\int_{1}^{\infty} g(x) , dx).

Analyze Convergence/Divergence of the Comparison Function: If the integral (\int_{1}^{\infty} g(x) , dx) converges, then by the comparison test, (\int_{1}^{\infty} f(x) , dx) also converges. Conversely, if (\int_{1}^{\infty} g(x) , dx) diverges, then ( \int_{1}^{\infty} f(x) , dx) diverges as well.
In the case of (f(x) = x^{x}), it's difficult to find a suitable function (g(x)) for comparison. Therefore, the integral test may not be directly applicable in this scenario. Other convergence tests such as the limit comparison test, ratio test, or root test might be more suitable for determining the convergence or divergence of this integral.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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