Broken Stick Model?

[narfenugen]

I'm supposed to use the broken stick model to get expected values for principle components (basically the same as the expected eigenvalue of the ith eigenvector).

Anyway the formulation I have for the formula is:

Sum of (1/(n-j)) from j=0 to n-i divided by n.

So when n = 3 here's what you get
1st eigenvalue --> i=1, [(1/3) + (1/2) + (1/1)] / 3 = .611
2nd eigenvalue --> i=2, [(1/3) + (1/2)] / 3 = .278
3rd eigenvalue--> i=3, (1/3)/3 = .111

Well that's all fine and dandy. But is there another formulation for this sum that makes calculating things easier when you've got a really large n? Or is there some fancy thing I missed back in the day that lets you add (1/1)+(1/2)+(1/3)+...+(1/n) really easily? I think I could figure it out from there. Any ideas?

I need to calculate the expected values of the first three eigenvalues for a case where my n is a couple thousand so I could really use some help on this one. I've got access to R if there's an easy way to make R calculate that summation.

Edit
So with help I got it numerically, but I'm still curious to know if there's an alternative formula for the summation.

Here's the R-code for when n=3:
rev(cumsum(1/(3:1)))/3