On 17/04/2015 04:23, David Wilson wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
The LHS is an entire function (analytic on the whole of the complex plane). To prove this it suffices that the sum of |1-cos(x/2^n)| be locally uniformly convergent, which it is because for x in any compact region the series looks like a convergent g.p. once n is large enough. It further follows that the only zeros of the LHS are those of its factors -- i.e., simple zeros at 0 (from the initial x) and at odd multiples of 2^n pi/2 (from cos x/2^n). So the LHS is an entire function with simple zeros at all integer multiples of pi and nowhere else. And so is the RHS. Therefore the ratio of the two sides is an entire function with no zeros and no poles. Or, to say that in simpler terms, a constant. If x is extremely small then the LHS is x + o(x). So is the RHS. So the constant ratio is in fact 1, and we're done. -- g