So, I am trying to verify the summation for the following formula, and then to figure out the average value of the formula.
So, if our $N = 3$, then the formula gives:
$\frac13*1+\frac23*(\frac23*2+\frac13*3*(1)) = {17\over 9}$
If $N=4$:
$\frac14*1+\frac34*(\frac24*2+\frac24(\frac34*3+\frac14*4*(1))) = {71\over 32}$
If $N=5$:
$\frac15*1+\frac45*(\frac25*2+\frac35*(\frac35*3+\frac25*(\frac45*4+\frac15*5*(1)))) = \frac{1569}{625}= 2.5104$
Now, the summation as I have figured it out is less a summation, and more of a method to get the total in terms of N. So, the starting portion is
${(n-1)^2\over n^2+1}$ Now, we go from i = 2 to n.
If i < n-1 then we multiply what we have by ${i\over n}$. If i < n-2, we add ${(n-i)^2\over n}$ When i=n, we add ${1\over n}$
This psuedocode above gets the right series of numbers.... but it's not a summation, which I do not need, but it would be nice to know how to do it. What I am really trying to figure out is given n alone, what is the value of the formula, in terms of n of course.