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

Printable View

- April 24th 2007, 06:09 PMclockinglyConvergence 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 - April 24th 2007, 07:33 PMThePerfectHacker
- April 24th 2007, 10:34 PMCaptainBlack
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