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

[Kristofer Munsterhjelm]

Raph Frank wrote:
> On Sat, May 22, 2010 at 11:39 PM,  <fsimmons at pcc.edu> wrote:
>> Satisfaction Approval Voting - A Better Proportional Representation Electoral Method
>>
>> One way to generalize Proportional Approval voting to range ballots is by
>> finding the most natural smooth extension of the function f that takes each
>> natural number n to the sum
>>
>> f(n) = 1 + 1/2 + ... + 1/n.
>>
>> It turns out that we can extend f(n) to all positive real values of n via the
>> integral
>>
>> Integral from zero to one of (1-t^n)/(1-t) with respect to t
> 
> Interesting.  The previous suggestion was to use logs and a correction.

That is probably because the integral approaches the natural logarithm 
as n goes to infinity. For integer values, f(n) is the nth harmonic 
number H_n, and

lim (n -> inf) H_n = ln(n) + gamma

where gamma is Euler's constant (0.5772...)

For instance, ln(10) + gamma = 2.879801 whereas H_10 is 2.828968.

That, in turn, is related to that the integral of 1/n is ln(n).