# How do you find the positive values of p for which #Sigma (1/n(lnn)^p)# from #[2,oo)# converges?

The series:

is convergent for

Consider the function:

or that:

In both cases by the squeeze theorem:

Now consider the derivative:

All the hypotheses of the integral test are then satisfied and the convergence of the series:

is equivalent to the convergence of the integral:

Clearly the integral is convergent only if:

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To find the positive values of ( p ) for which the series ( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^p} ) converges, you can use the integral test:

- Integrate the function ( f(x) = \frac{1}{x(\ln x)^p} ) from 2 to infinity.
- Determine the convergence of the improper integral.

If the integral converges, then the series also converges. So, find the values of ( p ) for which the integral converges.

To integrate ( f(x) ), you may need to use substitution or other integration techniques. After integrating, analyze the convergence of the integral.

If the integral converges, then the series ( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^p} ) converges. Therefore, the values of ( p ) for which the integral converges will be the values for which the series converges.

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To determine the positive values of ( p ) for which the series ( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^p} ) converges, you can use the integral test.

- Integrate the function ( \frac{1}{x(\ln x)^p} ) with respect to ( x ) from ( x = 2 ) to ( x = \infty ).
- Evaluate the integral.
- Determine for which values of ( p ) the integral converges.

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

Please note that finding the antiderivative of ( \frac{1}{x(\ln x)^p} ) and evaluating the integral may involve integration by parts or substitution, depending on the value of ( p ).

Once you've determined the integral and its convergence, you can find the values of ( p ) for which the series converges.

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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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- How do you determine if the improper integral converges or diverges #e^(-2t) dt # from negative infinity to -1?
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