Math Help - Convergence problem

1. Convergence problem

If anyone could explain how the following problem is done, it would be greatly appreciated.

Show that the following series either converges or diverges by using a suitable test:

sigma starting at j=2 and going to infinity of:
(-1)^j * ln(j)/j

2. Originally Posted by clockingly
If anyone could explain how the following problem is done, it would be greatly appreciated.

Show that the following series either converges or diverges by using a suitable test:

sigma starting at j=2 and going to infinity of:
(-1)^j * ln(j)/j
Use Leibniz Alternating Series Test.

3. Originally Posted by clockingly
If anyone could explain how the following problem is done, it would be greatly appreciated.

Show that the following series either converges or diverges by using a suitable test:

sigma starting at j=2 and going to infinity of:
(-1)^j * ln(j)/j
An alternating series converges if eventualy the absolute values of the terms
are monotonicaly decreasing with limit 0.

Consider:

f(x) = ln(x)/x

f'(x) = 1/x^2 - ln(x)/x^2

and for x>3 ln(x)>1, so f'(x)<0 for x>3 and so f(x) is decreasing for x>3.
So the absolute values of the terms of your series are decreasing for j>3.

Now it remains to show that lin_{j-> infty} ln(j)/j = 0, which can be done
using L'Hopital's rule on f(x)

RonL

RonL