# How do you use the integral test to determine if #1/(sqrt1(sqrt1+1))+1/(sqrt2(sqrt2+1))+1/(sqrt3(sqrt3+1))+...1/(sqrtn(sqrtn+1))+...# is convergent or divergent?

The series is divergent. See the explanation below.

This is the series:

Before starting with the integral test, we need to see that a couple conditions are met first.

Then:

This is just as valid a method as the integral test, but quicker.

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To use the integral test to determine the convergence or divergence of the series ( \frac{1}{\sqrt{1}(\sqrt{1}+1)} + \frac{1}{\sqrt{2}(\sqrt{2}+1)} + \frac{1}{\sqrt{3}(\sqrt{3}+1)} + \ldots + \frac{1}{\sqrt{n}(\sqrt{n}+1)} + \ldots ):

- Check if the series terms are positive, which they are.
- Verify that the series terms are decreasing, which they are.
- Integrate the function ( f(x) = \frac{1}{\sqrt{x}(\sqrt{x} + 1)} ) from ( x = 1 ) to ( x = \infty ).
- Determine if the integral converges or diverges.

If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

By integrating the function ( f(x) ) and evaluating the integral, you can determine the convergence or divergence of the given series.

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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.

- How do you apply the ratio test to determine if #sum_(n=1)^oo (e^n(n!))/n^n# is convergent or divergent?
- How do you determine if the improper integral converges or diverges #int sec^2 x dx# from negative 0 to pi?
- How do you use the integral test to determine the convergence or divergence of #1+1/sqrt2+1/sqrt3+1/sqrt4+...#?
- How do you test the improper integral #int (x(1+x^2)^-2)dx# from #[0,oo)# and evaluate if possible?
- How do you use the Integral test on the infinite series #sum_(n=1)^oo1/n^5# ?

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