For what values of p does the series converge?

[limonysal]

Homework Statement

For what values of p does the series converge?

Homework Equations

Σn=2 1/(n^p)(ln n)

The Attempt at a Solution

So far, all that is available to me is the integral test, the comparison test and the limit comparison test.

So using the comparison test. 1/(n^p)(ln n) < 1/n^p

1/n^p is a p series and only converges when p>1. So it converges.

But this doesn't feel right :(

 

[dynamicsolo]

So using the comparison test. 1/(n^p)(ln n) < 1/n^p

1/n^p is a p series and only converges when p>1. So it converges.

But this doesn't feel right :(

This is fine: the comparison of terms is valid for n > 1 (so the summation in the series would have to start at n = 2).

In having looked at a fair number of these p-series related problems, I've come to think of the contribution of a (ln n) factor as equivalent to adding zero to the p exponent when conducting the "p-test"; the natural logarithm in such series seems to make a negligible contribution to the convergence of the series.

 

[limonysal]

ah i meant to add in so it converges for p>1

its just that all the p series problems usually end up with p>1 XD

 

[dynamicsolo]

ah i meant to add in so it converges for p>1

its just that all the p series problems usually end up with p>1 XD

That's largely because of the problems authors tend to choose for courses making a first pass through material on infinite series. Not all p-series one may encounter turn out that way... ;-)