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:
By signing up, you agree to our Terms of Service and Privacy Policy
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.
By signing up, you agree to our Terms of Service and Privacy Policy
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.
By signing up, you agree to our Terms of Service and Privacy Policy
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.
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.
- Using the integral test, how do you show whether #sum 3/(n sqrt(ln(n)))# diverges or converges from n=1 to infinity?
- How do you determine whether the sequence #a_n=n/(ln(n)^2# converges, if so how do you find the limit?
- How do you determine if the series the converges conditionally, absolutely or diverges given #sum_(n=1)^oo (-1)^(n+1)arctan(n)#?
- How do you determine if the improper integral converges or diverges #e^(-2t) dt # from negative infinity to -1?
- How do you find #lim (3x^2+x+2)/(x-4)# as #x->0# using l'Hospital's Rule or otherwise?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7