I think, it oscillates. The solution oscillates between these 2 values, and both branches are present in it. GFR has many times tried to suggest this approach, use of both Lambert branches at the same time (2- valued function), or alternatively ( function implying hidden time variable).
The summation can be looked upon as behaviour of partial sums?
Like when You have series 1-1+1-1+1-1 - oscillating series, it has partial sums 1,0,1,0.....- also oscillating between 2 values, but value 1/2.
So the question is then what series give partial sums 1(/(1-log(2)) and 1/(1-2log(2))?
Obviosly, first one is 1+log(2)+(log(2)^2)+ log(2)^3- .........
Second is 1+ 2log(2)+ (2log(2))^2+ (2*( log(2)))^3 + = 1+log(4)-(log(4))^2 +(log(4))^3 .....
if we would use binary log instead of log base e , first series would become:
1+1+1+1+1 .......... with partial sums 1,2,3,4,5,6 with value -1/2.
Second:
1+2+4+8+16+32+64......with partial sums: 1,3, 7, 15, 31, 63, 127..........with value? (1/(1-2) )=-1 .
So the values of 2 series together oscillate between -1/2 and -1.
-1/2,-1,-1/2,-1 ......
The difference between terms of oscillations is 1/2 or = -1/2. second difference- +1 or -1 ; third difference: 2 or -2 etc.
Question is, shall we use log base e in the case of sqrt(2) when its natural to use base 2?
Admittedly, this post of mine is somewhat confusing.
Ivars
The summation can be looked upon as behaviour of partial sums?
Like when You have series 1-1+1-1+1-1 - oscillating series, it has partial sums 1,0,1,0.....- also oscillating between 2 values, but value 1/2.
So the question is then what series give partial sums 1(/(1-log(2)) and 1/(1-2log(2))?
Obviosly, first one is 1+log(2)+(log(2)^2)+ log(2)^3- .........
Second is 1+ 2log(2)+ (2log(2))^2+ (2*( log(2)))^3 + = 1+log(4)-(log(4))^2 +(log(4))^3 .....
if we would use binary log instead of log base e , first series would become:
1+1+1+1+1 .......... with partial sums 1,2,3,4,5,6 with value -1/2.
Second:
1+2+4+8+16+32+64......with partial sums: 1,3, 7, 15, 31, 63, 127..........with value? (1/(1-2) )=-1 .
So the values of 2 series together oscillate between -1/2 and -1.
-1/2,-1,-1/2,-1 ......
The difference between terms of oscillations is 1/2 or = -1/2. second difference- +1 or -1 ; third difference: 2 or -2 etc.
Question is, shall we use log base e in the case of sqrt(2) when its natural to use base 2?
Admittedly, this post of mine is somewhat confusing.
Ivars