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open problems / Discussion
#7
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
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Messages In This Thread
open problems / Discussion - by Gottfried - 05/30/2008, 07:08 AM
RE: open problems / Discussion - by bo198214 - 05/30/2008, 07:48 AM
RE: open problems / Discussion - by Ivars - 06/09/2008, 11:33 AM

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