# How do you test the series #(lnk)/(k^2)# for convergence?

The series:

is convergent.

To test the convergence of the series:

we can use the integral test, with test function:

Verify that the hypotheses of the integral test theorem are satisfied:

so the function is monotone decreasing in the interval

So the convergence of the series is equivalent to the convergence of the integral:

We start solving the indefinite integral by parts:

so:

The integral converges, so the series is also proven to be convergent.

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To test the series (\frac{\ln k}{k^2}) for convergence, you can use the integral test. Here's how:

- Integrate the function (f(x) = \frac{\ln x}{x^2}) from (1) to (n) to get (F(n)).
- If the integral (\int_1^\infty \frac{\ln x}{x^2} , dx) converges, then the series (\sum_{k=1}^\infty \frac{\ln k}{k^2}) converges. If the integral diverges, then the series diverges.

The integral test states that if (f(x)) is continuous, positive, and decreasing for (x \geq 1), then the convergence of the series (\sum_{k=1}^\infty f(k)) is equivalent to the convergence of the improper integral (\int_1^\infty f(x) , dx).

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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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