D: HALP GAIZ!

(Sorry if I'm not supposed to post homework here or something.)

My Analysis prof gives hard problems as extra-credit sometimes, and he gave this one during our first class of the semester yesterday.  I can't get anywhere with it. :(  It looks like it should be totally easy, but I'm not seeing it.  I just have to prove that the sequence {sin(1), sin(4), sin(9), ...} doesn't converge. The numbers 1,4,9... are the integer squares.

Anyone have any ideas?

Basically you want to show that the series n^2 is kind of independent of the series n*pi.

You want to find a infinite sequence of sqaures that are "close" to (2n+1/2)*pi and an infinite sequence that are "close" to (2n - 1/2)*pi.

By close enough I mean sufficiently close that sin of them is greater than 1/2 or less than -1/2.



I'd guess maybe you want to do it by using rational approximations to pi, and multiplying through by the denominator and numerator to find a multiple of pi "near" a square number.

Although, limitations to this, are I think the error of an approximation p/q is at best O(1/q^2) [this is a vague recollection], so you don't really get the multiple of pi very near the square number.




It's probably in fact easier to consider it in the abstract case, prove for an irrational number x and for e >0 that the sequence n*x comes with e of m^2 infinitely often, a statement which is invariably true.

Then tweak it a little so you can use (2n + 1/2)*x and (2n - 1/2)*x, then apply it to pi to get your result.


But I aint doing it for you