9
votes
2answers
320 views

Is there an elementary proof for Euler's product for Sine?

I've been looking at proofs of this equation: $$\displaystyle \frac {\sin \pi x} {\pi x} = \displaystyle \prod_{k \mathop = 1}^\infty \left({1 - \frac{x^2}{k^2} }\right)$$ All the proofs seem to ...
8
votes
6answers
978 views

Pointwise Convergence of $\sum \frac{\sin(\sqrt{n}x)}{n}$

I am having trouble in proving the pointwise convergence of $$ g(x)=\sum_{n=1}^\infty \frac{\sin(\sqrt{n}x)}{n}$$ for all real numbers $x$ using elementary methods (e.g. integral test, Weierstrass ...