Great xkcd today

flatmaster

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)

mathwonk

wouldn't that identity imply that pi is an algebraic number?

Hurkyl

Pro-tip: the following two phrases are not synonymous:

• not all of these are wrong
• all of these are right

Char. Limit

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.

Hurkyl

Char. Limit

Huh? I was talking about the cosine one.

Hurkyl

So was I. 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.

coolul007

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

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

DonAntonio

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

DonAntonio

jacobrhcp

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

SHISHKABOB

could I suggest linking to http://xkcd.com/1047/ instead, so that people on friday and so on don't become very confused reading this thread ;)

Borek

Done.