# Convergence problem

• Apr 24th 2007, 05:09 PM
clockingly
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
• Apr 24th 2007, 06:33 PM
ThePerfectHacker
Quote:

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.
• Apr 24th 2007, 09:34 PM
CaptainBlack
Quote:

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