[EM] EM] Satisfaction Approval Voting - A Better Proportional Representation Electoral Method

[Kristofer Munsterhjelm]

fsimmons at pcc.edu wrote:
> Note that the sum
> 
> 1+1/3+…+1/(2n+1)
> 
> is the integral (with respect to t) from zero to one of the sum
> 
> 1+t^2+…+t^(2n),
> 
> and that this integrand is a finite geometric sum with closed form
> 
> (1-t^(2n+1))/(1-t^2) .
> 
> So this is the appropriate integrand for a Sainte-Lague version that 
> allows fractional values of n, i.e. that works with any kind of range
> ballot, not just approval.

For the list, I'll note that the 2n+1 in the integrand appears to really 
be 2(n+1).

This gives

f(x) = H(x + 0.5)/2 + ln(2),

where H is the harmonic number function, so an approximation to H(x) can 
be used here as well.

(If the term had been 2n+1, then we'd have got f(x) = H(x)/2 + ln(2), 
which is not right.)