Mathematical fact about sums of sines (James)

[James Crook]

The mathematical fact I've chosen is:

The sum sin(x) + (sin(3x))/3 + (sin(5x))/5 +(sin(7x))/7 + ...

gives an increasingly good approximation to a 'square wave' the
further you go in the sum. Depending on x it can sum to pi/4, or
zero, or -pi/4. So for example, in degrees,

sin( 10 ) + (sin(30))/3 + (sin(50))/5 + (sin(70))/7 + ....

gets as close as we like to pi/4 - provided you go far enough. And so
does

sin( 11 ) + (sin(33))/3 + (sin(55))/5 + (sin(77))/7 + ....

And so does

sin( 9 ) + (sin(27))/3 + (sin(45))/5 + (sin(63))/7 + ....

And we get -pi/4 for x = -9, -10 and -11, and zero for x=0.