Series Riddle
1/1·2 + 1/2·4 + 1/3·8 + 1/4·16 + 1/5·32 + ... = ?
In ancapistan, if you have no property, you have no rights
Ancaps often declare, "All rights are property rights." I was thinking about this the other day, in the context of running into libertarians online who insisted that libertarianism supports "the freedom of movement," and realized that this principle actually entails that people without property have no rights at all, let alone any right to "freedom of movement." Of course, immediately, any ancap readers still left here are going to say, "Wait a second! Everyone owns his own body! And so everyone at least has the right to not have his body interfered with." Well, that is true... except that in ancapistan, one has no right to any place to put that body, except if one owns property, or has the permission of at least one property owner to place that body on her land. So, if one is landless and penniless, one had sure better hope that there are kindly disposed property owners aligned in a corridor from wherever one happens to be to wherever the...
17.0, but with an extra fraction that can never quite be positioned. It's there, but rounding causes it to go into infinity.
ReplyDeleteYou and I must interpret my admittedly clumsy typographic idiom differently from oneanother; the result can hardly overtop 1.
Delete= 2/1 + 4/2 + 8/3 + 16/4 + 32/5 ...
ReplyDelete= 2 + 2 + 2 2/3 + 4 + 6.4 + ...
Since each one gets larger, I'm going with infinity.
Or do you mean:
= 1/2 + 1/8 + 1/24 + 1/64 + 1/160... ?
Well, it pretty much stops growing after the first twenty or so terms, so 0.693147181...
But I have no idea how to do it analytically.
Sum [n=1:inf] (1 / (n * 2^n) )
Indeed it is Log(2), as the derivative of this series expansion is that of 1/(1-x). Also he doesn't know parentheses.
DeleteShiny I'm pretty sure Wabulon knows about parentheses. He's just lazy.
DeleteYou and I had the same mismatch as Fetz and I.
Deleteln(2).
ReplyDeleteAndy: one way to prove it is to write down the Taylor expansion at infinity of (1/x)*(1/(x-1)) and integrate it to see that the sum from 1 to infinity of 1/(n*x^n) is the Taylor series at infinity for ln(x/(x-1)). Evaluating at x=2 gives the result.
(Of course, it's slightly annoying/confusing that the Taylor series at infinity is really a Laurent series and not a power series at all.)
Yeah. And I do so know pearantesthes.
Deleteshonk,
ReplyDeleteGiven the information that it is ln(2), I can probably do the Taylor series expansion and then reduce it to the form above. What I don't know how to do is the inverse. Did you guys just recognize 0.693... as ln(2) and go from there? The only irrationals I'd have so recognized would be sqrt(2), 1/sqrt(2), e, Pi, Pi/2.
Yes, I definitely recognize 0.693... as ln(2) (for one thing, this is why the Rule of 70 works).
DeleteAs for working backwards, I immediately assumed that Wabulon's series was Laurent series with a variable in place of the 2. If you make that assumption then you immediately notice that the series gets nicer if you differentiate once, so that's what I did. What you get is practically the Taylor series at infinity for -1/(1-x) = 1/(x-1) (this is obvious if you've derived the Taylor series at 0 for 1/(1-x) as many times as I have); there's just an extra factor of 1/x. That gives that the derivative of Wabulon's series is the series for 1/(x(x-1)), so you just integrate to get the original function.
I did it your way; in fact, I chose the series to make that easy.
DeleteWhere is Ludwig Wittgenstein when you need him?
ReplyDeleteThe first few items in a series don't tell us "how to go on." - there are uncountably many ways of doing so!
But Kevin, surely Wittgenstein's point here Is that we do know how to go on, but that knowing is not itself a sort of rule or algorithm.
DeleteAbsolutely true, a point I made to my father as a child (he loved series riddles). Yet, as in love and war, despite all philosophy and logic in our way, somehow we know how to proceed.
Delete