Can you prove the identities for sqrt(2) and pi from the xkcd comic?

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Great xkcd today

http://xkcd.com/

At the bottom, he claims some identities expressing some roots in terms of pi.

Is he right? What's the easiest way to prove these identities?

Sqrt(2) = 3/5 - pi/(7-pi)
 
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wouldn't that identity imply that pi is an algebraic number?
 


flatmaster said:
At the bottom, he claims some identities expressing some roots in terms of pi.
Pro-tip: the following two phrases are not synonymous:
  • not all of these are wrong
  • all of these are right
 


Pro-tip: Only one of those "identities" is true. Bonus points if you figure out which it is.

I'm still trying to figure out HOW said identity is true. It's really kind of cool.
 


Char. Limit said:
Pro-tip: Only one of those "identities" is true. Bonus points if you figure out which it is.

I'm still trying to figure out HOW said identity is true. It's really kind of cool.
It's a geometric series, if you can figure out what the complex part of the numbers are supposed to be.
 


Hurkyl said:
It's a geometric series, if you can figure out what the complex part of the numbers are supposed to be.

Huh? I was talking about the cosine one.
 


Char. Limit said:
Huh? I was talking about the cosine one.
So was I. :biggrin: IIRC you can also work it out with induction and the sum-of-cosines formula, but the trick to turn it into a geometric series is faster. Of course, it's a bit annoying extracting the answer after you use the trick. There's another trick you can do that I think works out, but I haven't bothered fleshing it out.
cos x = Re{ exp(i x) }
 


flatmaster said:
Great xkcd today

http://xkcd.com/

At the bottom, he claims some identities expressing some roots in terms of pi.

Is he right? What's the easiest way to prove these identities?

Sqrt(2) = 3/5 - pi/(7-pi)

Squaring both sides results in the following equation which is false:

this is not even close
364 π+ 14 π^2 = 2009
 


coolul007 said:
Squaring both sides results in the following equation which is false:

this is not even close
364 π+ 14 π^2 = 2009



Yes, and mathwonk first noted it without the operations: it'd imply [itex]\pi[/itex] is rational, which is false, of course.

DonAntonio
 


So is there a simple identity for [itex]\sum_{n}n^{-n}[/itex]?