How do you use the integral test to determine the convergence or divergence of #1+1/root3(4)+1/root3(9)+1/root3(16)+1/root3(25)+...#?
The series:
is divergent.
We can write the series synthetically as:
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To use the integral test to determine the convergence or divergence of the series (1 + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{9}} + \frac{1}{\sqrt{16}} + \frac{1}{\sqrt{25}} + \dots), follow these steps:

Define the function (f(x)) to be the terms of the series: (f(x) = \frac{1}{\sqrt{x}}).

Verify that the function (f(x)) is continuous, positive, and decreasing for (x \geq 1), which is the domain relevant to our series.

Apply the integral test:
 Evaluate the improper integral ( \int_{1}^{\infty} f(x) , dx = \int_{1}^{\infty} \frac{1}{\sqrt{x}} , dx).

Integrate ( \int_{1}^{\infty} \frac{1}{\sqrt{x}} , dx): [ \int_{1}^{\infty} \frac{1}{\sqrt{x}} , dx = \lim_{b \to \infty} \left[2\sqrt{x}\right]{1}^{b} = \lim{b \to \infty} \left(2\sqrt{b}  2\right) ]

Determine the convergence of the integral:
 Since the limit exists and is finite, the integral converges.

Conclusion:
 By the integral test, since the integral converges, the original series (1 + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{9}} + \frac{1}{\sqrt{16}} + \frac{1}{\sqrt{25}} + \dots) also converges.
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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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